1 added
2 removed
Original
2026-01-01
Modified
2026-02-28
1
-
<p>227 Learners</p>
1
+
<p>264 Learners</p>
2
<p>Last updated on<strong>August 5, 2025</strong></p>
2
<p>Last updated on<strong>August 5, 2025</strong></p>
3
<p>We use the derivative of arctan(1/x), which is -1/(x² + 1), as a tool for understanding how the arctan(1/x) function changes with respect to a slight change in x. Derivatives help us calculate profit or loss in real-life situations. We will now discuss the derivative of arctan(1/x) in detail.</p>
3
<p>We use the derivative of arctan(1/x), which is -1/(x² + 1), as a tool for understanding how the arctan(1/x) function changes with respect to a slight change in x. Derivatives help us calculate profit or loss in real-life situations. We will now discuss the derivative of arctan(1/x) in detail.</p>
4
<h2>What is the Derivative of arctan(1/x)?</h2>
4
<h2>What is the Derivative of arctan(1/x)?</h2>
5
<p>We now understand the derivative of arctan(1/x). It is commonly represented as d/dx (arctan(1/x)) or (arctan(1/x))', and its value is -1/(x² + 1).</p>
5
<p>We now understand the derivative of arctan(1/x). It is commonly represented as d/dx (arctan(1/x)) or (arctan(1/x))', and its value is -1/(x² + 1).</p>
6
<p>The<a>function</a>arctan(1/x) has a clearly defined derivative, indicating it is differentiable within its domain.</p>
6
<p>The<a>function</a>arctan(1/x) has a clearly defined derivative, indicating it is differentiable within its domain.</p>
7
<p>The key concepts are mentioned below: Inverse Tangent Function: arctan(y) = x implies tan(x) = y.</p>
7
<p>The key concepts are mentioned below: Inverse Tangent Function: arctan(y) = x implies tan(x) = y.</p>
8
<p>Chain Rule: Rule for differentiating composite functions like arctan(1/x). Derivative of arctan(x): d/dx (arctan(x)) = 1/(1 + x²).</p>
8
<p>Chain Rule: Rule for differentiating composite functions like arctan(1/x). Derivative of arctan(x): d/dx (arctan(x)) = 1/(1 + x²).</p>
9
<h2>Derivative of arctan(1/x) Formula</h2>
9
<h2>Derivative of arctan(1/x) Formula</h2>
10
<p>The derivative of arctan(1/x) can be denoted as d/dx (arctan(1/x)) or (arctan(1/x))'.</p>
10
<p>The derivative of arctan(1/x) can be denoted as d/dx (arctan(1/x)) or (arctan(1/x))'.</p>
11
<p>The<a>formula</a>we use to differentiate arctan(1/x) is: d/dx (arctan(1/x)) = -1/(x² + 1)</p>
11
<p>The<a>formula</a>we use to differentiate arctan(1/x) is: d/dx (arctan(1/x)) = -1/(x² + 1)</p>
12
<p>The formula applies to all x where x ≠ 0.</p>
12
<p>The formula applies to all x where x ≠ 0.</p>
13
<h2>Proofs of the Derivative of arctan(1/x)</h2>
13
<h2>Proofs of the Derivative of arctan(1/x)</h2>
14
<p>We can derive the derivative of arctan(1/x) using proofs. To show this, we will use trigonometric identities along with the rules of differentiation.</p>
14
<p>We can derive the derivative of arctan(1/x) using proofs. To show this, we will use trigonometric identities along with the rules of differentiation.</p>
15
<p>There are several methods we use to prove this, such as: By First Principle Using Chain Rule Using Implicit Differentiation We will now demonstrate that the differentiation of arctan(1/x) results in -1/(x² + 1)</p>
15
<p>There are several methods we use to prove this, such as: By First Principle Using Chain Rule Using Implicit Differentiation We will now demonstrate that the differentiation of arctan(1/x) results in -1/(x² + 1)</p>
16
<p>using the above-mentioned methods: Using Chain Rule To prove the differentiation of arctan(1/x) using the chain rule, We use the formula: u = 1/x and f(u) = arctan(u)</p>
16
<p>using the above-mentioned methods: Using Chain Rule To prove the differentiation of arctan(1/x) using the chain rule, We use the formula: u = 1/x and f(u) = arctan(u)</p>
17
<p>Thus, y = arctan(1/x) = arctan(u) By chain rule: dy/dx = dy/du * du/dx We know dy/du = 1/(1 + u²) and du/dx = -1/x² Substitute u = 1/x to get: dy/dx = 1/(1 + (1/x)²) * (-1/x²) = -1/(x² + 1)</p>
17
<p>Thus, y = arctan(1/x) = arctan(u) By chain rule: dy/dx = dy/du * du/dx We know dy/du = 1/(1 + u²) and du/dx = -1/x² Substitute u = 1/x to get: dy/dx = 1/(1 + (1/x)²) * (-1/x²) = -1/(x² + 1)</p>
18
<p>Thus, the derivative is -1/(x² + 1). Using Implicit Differentiation Consider y = arctan(1/x).</p>
18
<p>Thus, the derivative is -1/(x² + 1). Using Implicit Differentiation Consider y = arctan(1/x).</p>
19
<p>Then, tan(y) = 1/x. Differentiating both sides with respect to x, using implicit differentiation: sec²(y) * dy/dx = -1/x² Since sec²(y) = 1 + tan²(y), substitute tan(y) = 1/x: (1 + (1/x)²) * dy/dx = -1/x² dy/dx = -1/(x² + 1)</p>
19
<p>Then, tan(y) = 1/x. Differentiating both sides with respect to x, using implicit differentiation: sec²(y) * dy/dx = -1/x² Since sec²(y) = 1 + tan²(y), substitute tan(y) = 1/x: (1 + (1/x)²) * dy/dx = -1/x² dy/dx = -1/(x² + 1)</p>
20
<p>Hence, proved.</p>
20
<p>Hence, proved.</p>
21
<h3>Explore Our Programs</h3>
21
<h3>Explore Our Programs</h3>
22
-
<p>No Courses Available</p>
23
<h2>Higher-Order Derivatives of arctan(1/x)</h2>
22
<h2>Higher-Order Derivatives of arctan(1/x)</h2>
24
<p>When a function is differentiated several times, the derivatives obtained are referred to as higher-order derivatives. Higher-order derivatives can be a little tricky.</p>
23
<p>When a function is differentiated several times, the derivatives obtained are referred to as higher-order derivatives. Higher-order derivatives can be a little tricky.</p>
25
<p>To understand them better, think of a car where the speed changes (first derivative) and the<a>rate</a>at which the speed changes (second derivative) also changes.</p>
24
<p>To understand them better, think of a car where the speed changes (first derivative) and the<a>rate</a>at which the speed changes (second derivative) also changes.</p>
26
<p>Higher-order derivatives make it easier to understand functions like arctan(1/x). For the first derivative of a function, we write f′(x), which indicates how the function changes or its slope at a certain point.</p>
25
<p>Higher-order derivatives make it easier to understand functions like arctan(1/x). For the first derivative of a function, we write f′(x), which indicates how the function changes or its slope at a certain point.</p>
27
<p>The second derivative is derived from the first derivative, which is denoted using f′′(x). Similarly, the third derivative, f′′′(x) is the result of the second derivative, and this pattern continues.</p>
26
<p>The second derivative is derived from the first derivative, which is denoted using f′′(x). Similarly, the third derivative, f′′′(x) is the result of the second derivative, and this pattern continues.</p>
28
<p>For the nth Derivative of arctan(1/x), we generally use fⁿ(x) for the nth derivative of a function f(x) which tells us the change in the rate of change. (continuing for higher-order derivatives).</p>
27
<p>For the nth Derivative of arctan(1/x), we generally use fⁿ(x) for the nth derivative of a function f(x) which tells us the change in the rate of change. (continuing for higher-order derivatives).</p>
29
<h2>Special Cases:</h2>
28
<h2>Special Cases:</h2>
30
<p>When x is 0, the derivative is undefined because arctan(1/x) has a vertical asymptote there. When x is 1, the derivative of arctan(1/x) = -1/(1² + 1) = -1/2.</p>
29
<p>When x is 0, the derivative is undefined because arctan(1/x) has a vertical asymptote there. When x is 1, the derivative of arctan(1/x) = -1/(1² + 1) = -1/2.</p>
31
<h2>Common Mistakes and How to Avoid Them in Derivatives of arctan(1/x)</h2>
30
<h2>Common Mistakes and How to Avoid Them in Derivatives of arctan(1/x)</h2>
32
<p>Students frequently make mistakes when differentiating arctan(1/x). These mistakes can be resolved by understanding the proper solutions. Here are a few common mistakes and ways to solve them:</p>
31
<p>Students frequently make mistakes when differentiating arctan(1/x). These mistakes can be resolved by understanding the proper solutions. Here are a few common mistakes and ways to solve them:</p>
33
<h3>Problem 1</h3>
32
<h3>Problem 1</h3>
34
<p>Calculate the derivative of arctan(1/x)·sin(x).</p>
33
<p>Calculate the derivative of arctan(1/x)·sin(x).</p>
35
<p>Okay, lets begin</p>
34
<p>Okay, lets begin</p>
36
<p>Here, we have f(x) = arctan(1/x)·sin(x).</p>
35
<p>Here, we have f(x) = arctan(1/x)·sin(x).</p>
37
<p>Using the product rule, f'(x) = u′v + uv′ In the given equation, u = arctan(1/x) and v = sin(x). Let’s differentiate each term, u′ = d/dx (arctan(1/x)) = -1/(x² + 1) v′ = d/dx (sin(x)) = cos(x)</p>
36
<p>Using the product rule, f'(x) = u′v + uv′ In the given equation, u = arctan(1/x) and v = sin(x). Let’s differentiate each term, u′ = d/dx (arctan(1/x)) = -1/(x² + 1) v′ = d/dx (sin(x)) = cos(x)</p>
38
<p>Substituting into the given equation, f'(x) = (-1/(x² + 1))·sin(x) + arctan(1/x)·cos(x)</p>
37
<p>Substituting into the given equation, f'(x) = (-1/(x² + 1))·sin(x) + arctan(1/x)·cos(x)</p>
39
<p>Thus, the derivative of the specified function is (-sin(x)/(x² + 1)) + arctan(1/x)·cos(x).</p>
38
<p>Thus, the derivative of the specified function is (-sin(x)/(x² + 1)) + arctan(1/x)·cos(x).</p>
40
<h3>Explanation</h3>
39
<h3>Explanation</h3>
41
<p>We find the derivative of the given function by dividing the function into two parts. The first step is finding its derivative and then combining them using the product rule to get the final result.</p>
40
<p>We find the derivative of the given function by dividing the function into two parts. The first step is finding its derivative and then combining them using the product rule to get the final result.</p>
42
<p>Well explained 👍</p>
41
<p>Well explained 👍</p>
43
<h3>Problem 2</h3>
42
<h3>Problem 2</h3>
44
<p>A company uses a sensor that measures the angle as arctan(1/x) for x meters away from the source. If x = 2 meters, find the rate of change of the angle.</p>
43
<p>A company uses a sensor that measures the angle as arctan(1/x) for x meters away from the source. If x = 2 meters, find the rate of change of the angle.</p>
45
<p>Okay, lets begin</p>
44
<p>Okay, lets begin</p>
46
<p>We have y = arctan(1/x) (rate of change of the angle)...(1)</p>
45
<p>We have y = arctan(1/x) (rate of change of the angle)...(1)</p>
47
<p>Now, we will differentiate the equation (1) Take the derivative of arctan(1/x): dy/dx = -1/(x² + 1)</p>
46
<p>Now, we will differentiate the equation (1) Take the derivative of arctan(1/x): dy/dx = -1/(x² + 1)</p>
48
<p>Given x = 2, substitute this into the derivative: dy/dx = -1/(2² + 1) = -1/5</p>
47
<p>Given x = 2, substitute this into the derivative: dy/dx = -1/(2² + 1) = -1/5</p>
49
<p>Hence, the rate of change of the angle at a distance x = 2 is -1/5.</p>
48
<p>Hence, the rate of change of the angle at a distance x = 2 is -1/5.</p>
50
<h3>Explanation</h3>
49
<h3>Explanation</h3>
51
<p>We find the rate of change of the angle at x = 2 as -1/5, meaning that for a small change in x near 2, the angle decreases at this rate.</p>
50
<p>We find the rate of change of the angle at x = 2 as -1/5, meaning that for a small change in x near 2, the angle decreases at this rate.</p>
52
<p>Well explained 👍</p>
51
<p>Well explained 👍</p>
53
<h3>Problem 3</h3>
52
<h3>Problem 3</h3>
54
<p>Derive the second derivative of the function y = arctan(1/x).</p>
53
<p>Derive the second derivative of the function y = arctan(1/x).</p>
55
<p>Okay, lets begin</p>
54
<p>Okay, lets begin</p>
56
<p>The first step is to find the first derivative, dy/dx = -1/(x² + 1)...(1)</p>
55
<p>The first step is to find the first derivative, dy/dx = -1/(x² + 1)...(1)</p>
57
<p>Now we will differentiate equation (1) to get the second derivative: d²y/dx² = d/dx [-1/(x² + 1)]</p>
56
<p>Now we will differentiate equation (1) to get the second derivative: d²y/dx² = d/dx [-1/(x² + 1)]</p>
58
<p>Using the quotient rule, d²y/dx² = [0 * (x² + 1) + 1 * 2x] / (x² + 1)² d²y/dx² = 2x/(x² + 1)²</p>
57
<p>Using the quotient rule, d²y/dx² = [0 * (x² + 1) + 1 * 2x] / (x² + 1)² d²y/dx² = 2x/(x² + 1)²</p>
59
<p>Therefore, the second derivative of the function y = arctan(1/x) is 2x/(x² + 1)².</p>
58
<p>Therefore, the second derivative of the function y = arctan(1/x) is 2x/(x² + 1)².</p>
60
<h3>Explanation</h3>
59
<h3>Explanation</h3>
61
<p>We use the step-by-step process, where we start with the first derivative.</p>
60
<p>We use the step-by-step process, where we start with the first derivative.</p>
62
<p>Using the quotient rule, we differentiate -1/(x² + 1). We then simplify the terms to find the final answer.</p>
61
<p>Using the quotient rule, we differentiate -1/(x² + 1). We then simplify the terms to find the final answer.</p>
63
<p>Well explained 👍</p>
62
<p>Well explained 👍</p>
64
<h3>Problem 4</h3>
63
<h3>Problem 4</h3>
65
<p>Prove: d/dx (arctan(1/x)²) = -2 * arctan(1/x)/(x² + 1).</p>
64
<p>Prove: d/dx (arctan(1/x)²) = -2 * arctan(1/x)/(x² + 1).</p>
66
<p>Okay, lets begin</p>
65
<p>Okay, lets begin</p>
67
<p>Let’s start using the chain rule: Consider y = (arctan(1/x))²</p>
66
<p>Let’s start using the chain rule: Consider y = (arctan(1/x))²</p>
68
<p>To differentiate, we use the chain rule: dy/dx = 2 * arctan(1/x) * d/dx [arctan(1/x)]</p>
67
<p>To differentiate, we use the chain rule: dy/dx = 2 * arctan(1/x) * d/dx [arctan(1/x)]</p>
69
<p>Since the derivative of arctan(1/x) is -1/(x² + 1), dy/dx = 2 * arctan(1/x) * (-1/(x² + 1))</p>
68
<p>Since the derivative of arctan(1/x) is -1/(x² + 1), dy/dx = 2 * arctan(1/x) * (-1/(x² + 1))</p>
70
<p>Substituting y = (arctan(1/x))², d/dx (arctan(1/x)²) = -2 * arctan(1/x)/(x² + 1) Hence proved.</p>
69
<p>Substituting y = (arctan(1/x))², d/dx (arctan(1/x)²) = -2 * arctan(1/x)/(x² + 1) Hence proved.</p>
71
<h3>Explanation</h3>
70
<h3>Explanation</h3>
72
<p>In this step-by-step process, we used the chain rule to differentiate the equation.</p>
71
<p>In this step-by-step process, we used the chain rule to differentiate the equation.</p>
73
<p>Then, we replace arctan(1/x) with its derivative.</p>
72
<p>Then, we replace arctan(1/x) with its derivative.</p>
74
<p>As a final step, we substitute y = (arctan(1/x))² to derive the equation.</p>
73
<p>As a final step, we substitute y = (arctan(1/x))² to derive the equation.</p>
75
<p>Well explained 👍</p>
74
<p>Well explained 👍</p>
76
<h3>Problem 5</h3>
75
<h3>Problem 5</h3>
77
<p>Solve: d/dx (arctan(1/x)/x).</p>
76
<p>Solve: d/dx (arctan(1/x)/x).</p>
78
<p>Okay, lets begin</p>
77
<p>Okay, lets begin</p>
79
<p>To differentiate the function, we use the quotient rule: d/dx (arctan(1/x)/x) = (d/dx (arctan(1/x)) * x - arctan(1/x) * d/dx(x))/x²</p>
78
<p>To differentiate the function, we use the quotient rule: d/dx (arctan(1/x)/x) = (d/dx (arctan(1/x)) * x - arctan(1/x) * d/dx(x))/x²</p>
80
<p>We will substitute d/dx (arctan(1/x)) = -1/(x² + 1) and d/dx(x) = 1 = (-1/(x² + 1) * x - arctan(1/x) * 1) / x² = (-x/(x² + 1) - arctan(1/x)) / x²</p>
79
<p>We will substitute d/dx (arctan(1/x)) = -1/(x² + 1) and d/dx(x) = 1 = (-1/(x² + 1) * x - arctan(1/x) * 1) / x² = (-x/(x² + 1) - arctan(1/x)) / x²</p>
81
<p>Therefore, d/dx (arctan(1/x)/x) = (-x/(x² + 1) - arctan(1/x)) / x²</p>
80
<p>Therefore, d/dx (arctan(1/x)/x) = (-x/(x² + 1) - arctan(1/x)) / x²</p>
82
<h3>Explanation</h3>
81
<h3>Explanation</h3>
83
<p>In this process, we differentiate the given function using the product rule and quotient rule. As a final step, we simplify the equation to obtain the final result.</p>
82
<p>In this process, we differentiate the given function using the product rule and quotient rule. As a final step, we simplify the equation to obtain the final result.</p>
84
<p>Well explained 👍</p>
83
<p>Well explained 👍</p>
85
<h2>FAQs on the Derivative of arctan(1/x)</h2>
84
<h2>FAQs on the Derivative of arctan(1/x)</h2>
86
<h3>1.Find the derivative of arctan(1/x).</h3>
85
<h3>1.Find the derivative of arctan(1/x).</h3>
87
<p>Using the chain rule for arctan(1/x) gives us: d/dx (arctan(1/x)) = -1/(x² + 1) (simplified)</p>
86
<p>Using the chain rule for arctan(1/x) gives us: d/dx (arctan(1/x)) = -1/(x² + 1) (simplified)</p>
88
<h3>2.Can we use the derivative of arctan(1/x) in real life?</h3>
87
<h3>2.Can we use the derivative of arctan(1/x) in real life?</h3>
89
<p>Yes, we can use the derivative of arctan(1/x) in real life to determine the rate of change of angles in various applications, such as in engineering, physics, and navigation.</p>
88
<p>Yes, we can use the derivative of arctan(1/x) in real life to determine the rate of change of angles in various applications, such as in engineering, physics, and navigation.</p>
90
<h3>3.Is it possible to take the derivative of arctan(1/x) at the point where x = 0?</h3>
89
<h3>3.Is it possible to take the derivative of arctan(1/x) at the point where x = 0?</h3>
91
<p>No, x = 0 is a point where arctan(1/x) is undefined, so it is impossible to take the derivative at this point (since the function does not exist there).</p>
90
<p>No, x = 0 is a point where arctan(1/x) is undefined, so it is impossible to take the derivative at this point (since the function does not exist there).</p>
92
<h3>4.What rule is used to differentiate arctan(1/x)/x?</h3>
91
<h3>4.What rule is used to differentiate arctan(1/x)/x?</h3>
93
<p>We use the<a>quotient</a>rule to differentiate arctan(1/x)/x, d/dx (arctan(1/x)/x) = (-x/(x² + 1) - arctan(1/x)) / x².</p>
92
<p>We use the<a>quotient</a>rule to differentiate arctan(1/x)/x, d/dx (arctan(1/x)/x) = (-x/(x² + 1) - arctan(1/x)) / x².</p>
94
<h3>5.Are the derivatives of arctan(1/x) and arctan(x) the same?</h3>
93
<h3>5.Are the derivatives of arctan(1/x) and arctan(x) the same?</h3>
95
<p>No, they are different. The derivative of arctan(1/x) is -1/(x² + 1), while the derivative of arctan(x) is 1/(1 + x²).</p>
94
<p>No, they are different. The derivative of arctan(1/x) is -1/(x² + 1), while the derivative of arctan(x) is 1/(1 + x²).</p>
96
<h2>Important Glossaries for the Derivative of arctan(1/x)</h2>
95
<h2>Important Glossaries for the Derivative of arctan(1/x)</h2>
97
<ul><li>Derivative: The derivative of a function indicates how the given function changes in response to a slight change in x.</li>
96
<ul><li>Derivative: The derivative of a function indicates how the given function changes in response to a slight change in x.</li>
98
</ul><ul><li>Inverse Tangent Function: The inverse of the tangent function, represented as arctan(x), gives the angle whose tangent is x.</li>
97
</ul><ul><li>Inverse Tangent Function: The inverse of the tangent function, represented as arctan(x), gives the angle whose tangent is x.</li>
99
</ul><ul><li>Chain Rule: A fundamental rule for differentiating composite functions, allowing us to find the derivative of functions like arctan(1/x).</li>
98
</ul><ul><li>Chain Rule: A fundamental rule for differentiating composite functions, allowing us to find the derivative of functions like arctan(1/x).</li>
100
</ul><ul><li>Quotient Rule: A method used to differentiate functions that are divided by one another, such as arctan(1/x)/x.</li>
99
</ul><ul><li>Quotient Rule: A method used to differentiate functions that are divided by one another, such as arctan(1/x)/x.</li>
101
</ul><ul><li>Asymptote: A line that a function approaches but never crosses, relevant for points where functions like arctan(1/x) become undefined.</li>
100
</ul><ul><li>Asymptote: A line that a function approaches but never crosses, relevant for points where functions like arctan(1/x) become undefined.</li>
102
</ul><p>What Is Calculus? 🔢 | Easy Tricks, Limits & 🎯 Fun Learning for Kids | ✨BrightCHAMPS Math</p>
101
</ul><p>What Is Calculus? 🔢 | Easy Tricks, Limits & 🎯 Fun Learning for Kids | ✨BrightCHAMPS Math</p>
103
<p>▶</p>
102
<p>▶</p>
104
<h2>Jaskaran Singh Saluja</h2>
103
<h2>Jaskaran Singh Saluja</h2>
105
<h3>About the Author</h3>
104
<h3>About the Author</h3>
106
<p>Jaskaran Singh Saluja is a math wizard with nearly three years of experience as a math teacher. His expertise is in algebra, so he can make algebra classes interesting by turning tricky equations into simple puzzles.</p>
105
<p>Jaskaran Singh Saluja is a math wizard with nearly three years of experience as a math teacher. His expertise is in algebra, so he can make algebra classes interesting by turning tricky equations into simple puzzles.</p>
107
<h3>Fun Fact</h3>
106
<h3>Fun Fact</h3>
108
<p>: He loves to play the quiz with kids through algebra to make kids love it.</p>
107
<p>: He loves to play the quiz with kids through algebra to make kids love it.</p>