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${:[h(x)=(5-6x)^(5)],[h^(')(x)=?]:} Choose 1 answer: (A) -30(5-6x)^(4) (B) (-6)^(5) (C) -6x^(5)+5x^(4)(5-6x) (D) 5(5-6x)^(4)$

$\begin{aligned} h(x) & =(5-6 x)^{5} \\ h^{\prime}(x) & =? \end{aligned}$ $\newline$ Choose $1$ answer: $\newline$ (A) $-30(5-6 x)^{4}$ $\newline$ (B) $(-6)^{5}$ $\newline$ (C) $-6 x^{5}+5 x^{4}(5-6 x)$ $\newline$ (D) $5(5-6 x)^{4}$

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Find derivatives of sine and cosine functions

Full solution

Q.

\begin{aligned} h(x) & =(5-6 x)^{5} \\ h^{\prime}(x) & =? \end{aligned}

\newline

Choose

1

answer:

\newline

(A)

-30(5-6 x)^{4}

\newline

(B)

(-6)^{5}

\newline

(C)

-6 x^{5}+5 x^{4}(5-6 x)

\newline

(D)

5(5-6 x)^{4}

Apply Power and Chain Rule: Apply the power rule combined with the chain rule to differentiate $h(x)$ . The power rule states that the derivative of $x^n$ is $n\cdot x^{(n-1)}$ . The chain rule states that the derivative of a composite function $f(g(x))$ is $f'(g(x))\cdot g'(x)$ . In this case, we have a composite function where the outer function is $u^5$ and the inner function is $u = 5 - 6x$ . We will first take the derivative of the outer function with respect to $u$ and then multiply it by the derivative of the inner function with respect to $x$ .
Differentiate Outer Function: Differentiate the outer function with respect to $u$ . The derivative of $u^5$ with respect to $u$ is $5u^{5-1} = 5u^4$ .
Differentiate Inner Function: Differentiate the inner function with respect to $x$ . The derivative of $5 - 6x$ with respect to $x$ is $-6$ .
Apply Chain Rule: Apply the chain rule by multiplying the derivatives from Step $2$ and Step $3$ . $\newline$ $h'(x) = 5\cdot(5 - 6x)^4 \cdot (-6)$
Simplify Expression: Simplify the expression. $h'(x) = -30(5 - 6x)^4$

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