Derivative of Root 3x
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Last updated on August 5, 2025

The derivative of root 3x, expressed as (3x)^(1/2), is used to analyze how this function changes in response to small changes in x. Derivatives are crucial in calculating rates of change in various real-life situations. We will now discuss the derivative of root 3x in detail.

What is the Derivative of Root 3x?

The derivative of root 3x is commonly represented as d/dx [(3x)^(1/2)] or [(3x)^(1/2)]', and its value is (3/2)(3x)^(-1/2). The function (3x)^(1/2) has a clearly defined derivative, indicating it is differentiable within its domain. The key concepts are mentioned below: Square Root Function: ((3x)^(1/2) indicates the square root of 3x). Power Rule: Rule for differentiating (3x)^(1/2).

Derivative of Root 3x Formula

The derivative of root 3x can be denoted as d/dx [(3x)^(1/2)] or [(3x)^(1/2)]'. The formula used to differentiate root 3x is: d/dx [(3x)^(1/2)] = (3/2)(3x)^(-1/2) The formula applies to all x where x > 0.

Proofs of the Derivative of Root 3x

We can derive the derivative of root 3x using proofs. To show this, we will use the rules of differentiation. There are several methods we use to prove this, such as: By First Principle Using Chain Rule Using the Power Rule We will now demonstrate that the differentiation of (3x)^(1/2) results in (3/2)(3x)^(-1/2) using the above-mentioned methods: By First Principle The derivative of (3x)^(1/2) can be proved using the First Principle, which expresses the derivative as the limit of the difference quotient. To find the derivative of (3x)^(1/2) using the first principle, we will consider f(x) = (3x)^(1/2). Its derivative can be expressed as the following limit. f'(x) = limₕ→₀ [f(x + h) - f(x)] / h … (1) Given that f(x) = (3x)^(1/2), we write f(x + h) = (3(x + h))^(1/2). Substituting these into equation (1), f'(x) = limₕ→₀ [(3(x + h))^(1/2) - (3x)^(1/2)] / h Using binomial expansion and simplification, f'(x) = (3/2)(3x)^(-1/2). Hence, proved. Using Chain Rule To prove the differentiation of (3x)^(1/2) using the chain rule, Let u = 3x, then f(x) = u^(1/2). By the chain rule, d/dx [u^(1/2)] = (1/2)u^(-1/2) * (du/dx). Since du/dx = 3, d/dx [(3x)^(1/2)] = (1/2)(3x)^(-1/2) * 3 = (3/2)(3x)^(-1/2). Using the Power Rule We will now prove the derivative of (3x)^(1/2) using the power rule. f(x) = (3x)^(1/2) can be expressed as (3x)^0.5. Using the power rule, d/dx [u^n] = n*u^(n-1) * (du/dx). We have u = 3x, n = 0.5, and du/dx = 3. d/dx [(3x)^(1/2)] = 0.5*(3x)^(-0.5) * 3 = (3/2)(3x)^(-1/2).

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Higher-Order Derivatives of Root 3x

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 (3x)^(1/2). 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 (3x)^(1/2), 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).

Special Cases:

When x = 0, the derivative is undefined because (3x)^(1/2) is not defined for x = 0. When x = 1, the derivative of (3x)^(1/2) is (3/2)(3)^(1/2).

Common Mistakes and How to Avoid Them in Derivatives of Root 3x

Students frequently make mistakes when differentiating (3x)^(1/2). 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 ((3x)^(1/2) * x^2)

Okay, lets begin

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

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.

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Problem 2

A company is analyzing the growth of its plant species, modeled by the function y = (3x)^(1/2), where y represents the height of the plant at time x. If x = 4 weeks, measure the growth rate of the plant.

Okay, lets begin

We have y = (3x)^(1/2) (growth rate of the plant)...(1) Now, we will differentiate the equation (1) Take the derivative: dy/dx = (3/2)(3x)^(-1/2) Given x = 4 (substitute this into the derivative), dy/dx = (3/2)(3*4)^(-1/2) = (3/2)(12)^(-1/2) = (3/2)(1/√12) = (3/2)(1/2√3) = 3/(4√3) Hence, the growth rate of the plant at x = 4 weeks is 3/(4√3).

Explanation

We find the growth rate of the plant at x = 4 weeks as 3/(4√3), which means that at a given time, the height of the plant would increase at this rate.

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Problem 3

Derive the second derivative of the function y = (3x)^(1/2).

Okay, lets begin

The first step is to find the first derivative, dy/dx = (3/2)(3x)^(-1/2)...(1) Now we will differentiate equation (1) to get the second derivative: d²y/dx² = d/dx [(3/2)(3x)^(-1/2)] = (3/2)(-1/2)(3x)^(-3/2) * 3 = -(9/4)(3x)^(-3/2). Therefore, the second derivative of the function y = (3x)^(1/2) is -(9/4)(3x)^(-3/2).

Explanation

We use the step-by-step process, where we start with the first derivative. Using the power rule, we differentiate (3x)^(-1/2). We then substitute the values and simplify the terms to find the final answer.

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Problem 4

Prove: d/dx [(3x)^(3/2)] = (9/2)x^(1/2).

Okay, lets begin

Let’s start using the chain rule: Consider y = (3x)^(3/2) To differentiate, we use the power rule: dy/dx = (3/2)(3x)^(1/2) * (3) = (9/2)x^(1/2) Substituting y = (3x)^(3/2), d/dx [(3x)^(3/2)] = (9/2)x^(1/2) Hence proved.

Explanation

In this step-by-step process, we used the power rule to differentiate the equation. Then, we replace (3x)^(3/2) with its derivative. As a final step, we substitute y = (3x)^(3/2) to derive the equation.

Well explained 👍

Problem 5

Solve: d/dx [(3x)^(1/2)/x]

Okay, lets begin

To differentiate the function, we use the quotient rule: d/dx [(3x)^(1/2)/x] = (d/dx [(3x)^(1/2)].x - (3x)^(1/2).d/dx(x)) / x² We will substitute d/dx [(3x)^(1/2)] = (3/2)(3x)^(-1/2) and d/dx(x) = 1 = [(3/2)(3x)^(-1/2).x - (3x)^(1/2)] / x² = [(3/2)x(3x)^(-1/2) - (3x)^(1/2)] / x² = [(3/2)(x/sqrt(3x)) - sqrt(3x)] / x² = [3/(2sqrt(3x)) - sqrt(3x)] / x². Therefore, d/dx [(3x)^(1/2)/x] = [3/(2sqrt(3x)) - sqrt(3x)] / x².

Explanation

In this process, we differentiate the given function using the quotient rule. As a final step, we simplify the equation to obtain the final result.

Well explained 👍

FAQs on the Derivative of Root 3x

1.Find the derivative of root 3x.

Using the power rule, d/dx [(3x)^(1/2)] = (3/2)(3x)^(-1/2).

2.Can we use the derivative of root 3x in real life?

Yes, we can use the derivative of root 3x in real life, such as in calculating the rate of growth or decay in biological and economic models.

3.Is it possible to take the derivative of root 3x at the point where x = 0?

No, x = 0 is a point where root 3x is undefined, so it is impossible to take the derivative at these points (since the function does not exist there).

4.What rule is used to differentiate (3x)^(1/2)/x?

We use the quotient rule to differentiate (3x)^(1/2)/x, d/dx [(3x)^(1/2)/x] = [(x.d/dx (3x)^(1/2) - (3x)^(1/2).1) / x²].

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

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

6.Can we find the derivative of the (3x)^(1/2) formula?

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

Important Glossaries for the Derivative of Root 3x

Derivative: The derivative of a function indicates how the given function changes in response to a slight change in x. Square Root Function: A function involving the square root of a variable, such as (3x)^(1/2). Power Rule: A basic rule for differentiation used for functions in the form of xⁿ, where n is a real number. Chain Rule: A differentiation rule used to differentiate the composition of functions. Higher-Order Derivative: The result of differentiating a function multiple times, giving insights into its changing rates.

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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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