What is the value of (d)/(dx)(x^((1)/(3))) at x=8 ?

Identify function: Identify the function to differentiate. We need to find the derivative of the function f(x) = x^(1/3) with respect to x.Apply power rule: Apply the power rule for differentiation. The power rule states that the derivative of x^n with respect to x is n*x^(n-1). In this case, n = 1/3, so the derivative of x^(1/3) is (1/3)*x^(1/3 - 1) = (1/3)*x^(-2/3).Simplify derivative: Simplify the expression for the derivative. The derivative simplifies to f'(x) = (1/3)*x^(-2/3).Evaluate at x=8: Evaluate the derivative at x=8. Substitute x with 8 in the derivative to get f'(8) = (1/3)*8^(-2/3).Calculate 8^(-2/3): Calculate 8^(-2/3). Since 8 = 2^3, we can rewrite 8^(-2/3) as (2^3)^(-2/3) = 2^(-2) = 1/(2^2) = 1/4.Multiply coefficient: Multiply the coefficient by the result from Step 5. Now, f'(8) = (1/3)*(1/4) = 1/12.

Resources

Testimonials

Plans

Resources

Testimonials

Plans

AI tutor

Welcome to Bytelearn!

Let’s check out your problem:

What is the value of
(d)/(dx)(x^((1)/(3))) at
x=8 ?

What is the value of $\frac{d}{d x}\left(x^{\frac{1}{3}}\right)$ at $x=8$ ?

View step-by-step help

Home

Math Problems

Precalculus

Evaluate rational exponents

Full solution

Q. What is the value of

\frac{d}{d x}\left(x^{\frac{1}{3}}\right)

x=8

Identify function: Identify the function to differentiate. $\newline$ We need to find the derivative of the function $f(x) = x^{\frac{1}{3}}$ with respect to $x$ .
Apply power rule: Apply the power rule for differentiation. $\newline$ The power rule states that the derivative of $x^n$ with respect to $x$ is $n*x^{(n-1)}$ . In this case, $n = \frac{1}{3}$ , so the derivative of $x^{\frac{1}{3}}$ is $(\frac{1}{3})*x^{(\frac{1}{3} - 1)} = (\frac{1}{3})*x^{-\frac{2}{3}}$ .
Simplify derivative: Simplify the expression for the derivative. $\newline$ The derivative simplifies to $f'(x) = (\frac{1}{3})\cdot x^{(-\frac{2}{3})}$ .
Evaluate at $x=8$ : Evaluate the derivative at $x=8$ . $\newline$ Substitute $x$ with $8$ in the derivative to get $f'(8) = (\frac{1}{3})\cdot8^{(-\frac{2}{3})}$ .
Calculate $8^{(-2/3)}$ : Calculate $8^{(-2/3)}$ . $\newline$ Since $8 = 2^3$ , we can rewrite $8^{(-2/3)}$ as $(2^3)^{(-2/3)} = 2^{-2} = 1/(2^2) = 1/4$ .
Multiply coefficient: Multiply the coefficient by the result from Step $5$ . $\newline$ Now, $f'(8) = (\frac{1}{3})*(\frac{1}{4}) = \frac{1}{12}.$