Given the function f(x)=(-9x-5)^((1)/(3)), find f^(')(x) in any form.

Identify function: Identify the function to differentiate. We are given the function f(x) = (-9x - 5)^(1/3), which is a composition of functions. We need to find its derivative f'(x).Apply chain rule: Apply the chain rule for differentiation. The chain rule states that the derivative of a composite function is the derivative of the outer function evaluated at the inner function times the derivative of the inner function. In this case, the outer function is u^(1/3) and the inner function is u = -9x - 5.Differentiate outer function: Differentiate the outer function with respect to the inner function. Let u = -9x - 5. Then the outer function is u^(1/3). The derivative of u^(1/3) with respect to u is (1/3)u^(-2/3).Differentiate inner function: Differentiate the inner function with respect to x. The derivative of u = -9x - 5 with respect to x is du/dx = -9.Apply chain rule multiplication: Apply the chain rule by multiplying the derivatives from Step 3 and Step 4. f'(x) = (d/dx of the outer function evaluated at the inner function) * (d/dx of the inner function) f'(x) = (1/3)(-9x - 5)^(-2/3) * (-9)Simplify derivative: Simplify the expression for the derivative. f'(x) = -3(-9x - 5)^(-2/3)

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Given the function $\newline$ $f(x)=(-9x-5)^{\frac{1}{3}},$ find $\newline$ $f'(x)$ in any form.

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Q. Given the function

\newline

f(x)=(-9x-5)^{\frac{1}{3}},

find

\newline

f'(x)

in any form.

Identify function: Identify the function to differentiate. $\newline$ We are given the function $f(x) = (-9x - 5)^{\frac{1}{3}}$ , which is a composition of functions. We need to find its derivative $f'(x)$ .
Apply chain rule: Apply the chain rule for differentiation. $\newline$ The chain rule states that the derivative of a composite function is the derivative of the outer function evaluated at the inner function times the derivative of the inner function. In this case, the outer function is $u^{(1/3)}$ and the inner function is $u = -9x - 5$ .
Differentiate outer function: Differentiate the outer function with respect to the inner function. $\newline$ Let $u = -9x - 5$ . Then the outer function is $u^{(1/3)}$ . The derivative of $u^{(1/3)}$ with respect to $u$ is $(1/3)u^{(-2/3)}$ .
Differentiate inner function: Differentiate the inner function with respect to $x$ . The derivative of $u = -9x - 5$ with respect to $x$ is $\frac{du}{dx} = -9$ .
Apply chain rule multiplication: Apply the chain rule by multiplying the derivatives from Step $3$ and Step $4$ . $\newline$ f'(x) = \left(\frac{d}{dx} of the outer function evaluated at the inner function\) $\times$ \left(\frac{d}{dx} of the inner function\) $\newline$ $f'(x) = \left(\frac{1}{3}\right)\left(-9x - 5\right)^{-\frac{2}{3}} \times (-9)$
Simplify derivative: Simplify the expression for the derivative. $f'(x) = -3(-9x - 5)^{-\frac{2}{3}}$