Derivative of x^3/2
2026-02-28 10:08 Diff
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Last updated on October 10, 2025

We use the derivative of x^(3/2), which is (3/2)x^(1/2), as a measuring tool for how the function changes in response to a slight change in x. Derivatives help us calculate profit or loss in real-life situations. We will now talk about the derivative of x^(3/2) in detail.

What is the Derivative of x^3/2?

We now understand the derivative of x^(3/2). It is commonly represented as d/dx (x^(3/2)) or (x^(3/2))', and its value is (3/2)x^(1/2). The function x^(3/2) has a clearly defined derivative, indicating it is differentiable within its domain.

The key concepts are mentioned below:

Power Function: (x(3/2)).

Power Rule: Rule for differentiating functions of the form xn.

Square Root Function: Results in the derivative having a square root form.

Derivative of x^3/2 Formula

The derivative of x(3/2) can be denoted as d/dx (x(3/2)) or (x(3/2))'.

The formula we use to differentiate x(3/2) is: d/dx (x(3/2)) = (3/2)x(1/2) (or) (x(3/2))' = (3/2)x(1/2)

The formula applies to all x where x > 0.

Proofs of the Derivative of x^3/2

We can derive the derivative of x^(3/2) using proofs. To show this, we will use the power rule along with basic differentiation rules.

There are several methods we use to prove this, such as:

  • By First Principle
     
  • Using Power Rule
     
  • Using Chain Rule

We will now demonstrate that the differentiation of x(3/2) results in (3/2)x(1/2) using the above-mentioned methods:

By First Principle

The derivative of x(3/2) can be proved using the First Principle, which expresses the derivative as the limit of the difference quotient. To find the derivative of x(3/2) using the first principle, we will consider f(x) = x(3/2). Its derivative can be expressed as the following limit. f'(x) = limₕ→₀ [f(x + h) - f(x)] / h … (1) Given that f(x) = x(3/2), we write f(x + h) = (x + h)(3/2). Substituting these into equation (1), f'(x) = limₕ→₀ [(x + h)(3/2) - x(3/2)] / h = limₕ→₀ [(x(3/2) + (3/2)x(1/2)h + ...) - x(3/2)] / h = limₕ→₀ [(3/2)x(1/2)h + ...] / h Simplifying the terms: f'(x) = limₕ→₀ (3/2)x(1/2) + ... As h approaches 0, the higher-order terms vanish, leaving: f'(x) = (3/2)x(1/2) Hence, proved.

Using Power Rule

To prove the differentiation of x(3/2) using the power rule, We use the formula: d/dx (x^n) = nx^(n-1) For x^(3/2), let n = 3/2. So, d/dx (x^(3/2)) = (3/2)x(3/2 - 1) = (3/2)x^(1/2) Thus, using the power rule, we get the derivative as (3/2)x^(1/2).

Using Chain Rule

We will now prove the derivative of x(3/2) using the chain rule. The step-by-step process is demonstrated below: Here, we use the expression: x(3/2) = (x1)(3/2) Let u = x, then u(3/2) = x^(3/2) The derivative using the chain rule: d/dx (u(3/2)) = (3/2)u(1/2) * du/dx Since u = x, du/dx = 1 Therefore, d/dx (x(3/2)) = (3/2)x(1/2) Thus, the derivative is (3/2)x(1/2).

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Higher-Order Derivatives of x^3/2

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. To understand them better, think of a car where the speed changes (first derivative) and the rate at which the speed changes (second derivative) also changes. Higher-order derivatives make it easier to understand functions like x^(3/2).

For the first derivative of a function, we write f′(x), which indicates how the function changes or its slope at a certain point. 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.

For the nth Derivative of x(3/2), we generally use f n(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).

Special Cases:

When x is 0, the derivative is not defined because x(3/2) is not differentiable at x = 0.

For positive values of x, the derivative of x(3/2) is positive, indicating an increasing function.

Common Mistakes and How to Avoid Them in Derivatives of x^3/2

Students frequently make mistakes when differentiating x^(3/2). These mistakes can be resolved by understanding the proper solutions. Here are a few common mistakes and ways to solve them:

Problem 1

Calculate the derivative of (x^(3/2) * x)

Okay, lets begin

Here, we have f(x) = x^(3/2) * x. Using the product rule, f'(x) = u′v + uv′ In the given equation, u = x^(3/2) and v = x. Let’s differentiate each term, u′= d/dx (x^(3/2)) = (3/2)x^(1/2) v′= d/dx (x) = 1 Substituting into the given equation, f'(x) = ((3/2)x^(1/2)). x + (x^(3/2)). 1 Let’s simplify terms to get the final answer, f'(x) = (3/2)x^(3/2) + x^(3/2) Thus, the derivative of the specified function is (5/2)x^(3/2).

Explanation

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.

Well explained 👍

Problem 2

XYZ Construction is designing a ramp where the height is represented by the function y = x^(3/2), where y is the elevation at a distance x. If x = 4 meters, determine the rate of height increase at that point.

Okay, lets begin

We have y = x^(3/2) (height of the ramp)...(1) Now, we will differentiate the equation (1) Take the derivative of x^(3/2): dy/dx = (3/2)x^(1/2) Given x = 4 (substitute this into the derivative) dy/dx = (3/2)*4^(1/2) dy/dx = (3/2)*2 = 3 Hence, we get the rate of height increase at a distance x = 4 meters as 3.

Explanation

We find the rate of height increase at x=4 as 3, which means that at a given point, the height increases by 3 meters per unit increase in horizontal distance.

Well explained 👍

Problem 3

Derive the second derivative of the function y = x^(3/2).

Okay, lets begin

The first step is to find the first derivative, dy/dx = (3/2)x^(1/2)...(1) Now we will differentiate equation (1) to get the second derivative: d2y/dx2 = d/dx [(3/2)x^(1/2)] Here we use the power rule, d2y/dx2 = (3/2) * (1/2)x^(-1/2) = (3/4)x^(-1/2) Therefore, the second derivative of the function y = x^(3/2) is (3/4)x^(-1/2).

Explanation

We use the step-by-step process, where we start with the first derivative.

Using the power rule, we differentiate (3/2)x^(1/2).

We then simplify the terms to find the final answer.

Well explained 👍

Problem 4

Prove: d/dx ((x + 2)^(3/2)) = 3(x + 2)^(1/2).

Okay, lets begin

Let’s start using the chain rule: Consider y = (x + 2)^(3/2) To differentiate, we use the chain rule: dy/dx = (3/2)(x + 2)^(1/2) * d/dx (x + 2) Since the derivative of (x + 2) is 1, dy/dx = (3/2)(x + 2)^(1/2) Hence, proved.

Explanation

In this step-by-step process, we used the chain rule to differentiate the equation.

Then, we replace (x + 2) with its derivative.

As a final step, we derive the equation.

Well explained 👍

Problem 5

Solve: d/dx ((x^(3/2))/x)

Okay, lets begin

To differentiate the function, we simplify it first: d/dx ((x^(3/2))/x) = d/dx (x^(1/2)) Using the power rule: d/dx (x^(1/2)) = (1/2)x^(-1/2) Therefore, d/dx ((x^(3/2))/x) = (1/2)x^(-1/2)

Explanation

In this process, we simplify the given function before differentiation.

Using the power rule, we find the derivative of the simplified function.

Well explained 👍

FAQs on the Derivative of x^3/2

1.Find the derivative of x^(3/2).

Using the power rule for x^(n), we have: d/dx (x^(3/2)) = (3/2)x^(1/2).

2.Can we use the derivative of x^(3/2) in real life?

Yes, we can use the derivative of x^(3/2) in real life in calculating rates of change, especially in fields such as mathematics, physics, and engineering.

3.Is it possible to take the derivative of x^(3/2) at the point where x = 0?

No, x = 0 is a point where x^(3/2) is not differentiable for real numbers, so it is impossible to take the derivative at this point.

4.What rule is used to differentiate (x^3/2) * x?

We use the product rule to differentiate (x^(3/2) * x): d/dx (x^(3/2) * x) = x * (3/2)x^(1/2) + x^(3/2) * 1.

5.Are the derivatives of x^(3/2) and (x^(3/2))^-1 the same?

No, they are different. The derivative of x^(3/2) is (3/2)x^(1/2), while the derivative of (x^(3/2))^-1 is -((3/2)x^(1/2))/x^3.

6.Can we find the derivative of the x^(3/2) formula?

To find, consider y = x^(3/2). Using the power rule: y’ = (3/2)x^(1/2).

Important Glossaries for the Derivative of x^3/2

  • Derivative: The derivative of a function indicates how the given function changes in response to a slight change in x.
  • Power Rule: A fundamental rule in calculus used to differentiate functions of the form x^n.
  • Chain Rule: A method in calculus for finding the derivative of the composition of two or more functions.
  • Product Rule: A rule used to find the derivative of the product of two functions.
  • Square Root Function: A function that involves the square root of a variable, often appearing in derivatives of fractional powers. ```

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Jaskaran Singh Saluja

About the Author

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.

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: He loves to play the quiz with kids through algebra to make kids love it.