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

Identify Functions: To find the derivative of the function h(x) = (5 - 6x)^5, we will use the chain rule. 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.Find Derivatives: First, let's identify the outer function and the inner function. The outer function is f(u) = u^5 and the inner function is u(x) = 5 - 6x. We need to find the derivatives of both functions.Apply Chain Rule: The derivative of the outer function f(u) = u^5 with respect to u is f'(u) = 5u^4.Substitute Derivatives: The derivative of the inner function u(x) = 5 - 6x with respect to x is u'(x) = -6.Simplify Expression: Now, we apply the chain rule: h'(x) = f'(u(x)) * u'(x). Substituting the derivatives we found, we get h'(x) = 5(5 - 6x)^4 * (-6).Simplify the expression by multiplying the constants: h'(x) = -30(5 - 6x)^4.

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Math Problems

Calculus

Find higher derivatives of rational and radical functions

Full solution

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

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Choose

1

answer:

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(A)

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

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(B)

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

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(C)

(-6)^{5}

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(D)

5(5-6 x)^{4}

Identify Functions: To find the derivative of the function $h(x) = (5 - 6x)^5$ , we will use the chain rule. 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.
Find Derivatives: First, let's identify the outer function and the inner function. The outer function is $f(u) = u^5$ and the inner function is $u(x) = 5 - 6x$ . We need to find the derivatives of both functions.
Apply Chain Rule: The derivative of the outer function $f(u) = u^5$ with respect to $u$ is $f'(u) = 5u^4$ .
Substitute Derivatives: The derivative of the inner function $u(x) = 5 - 6x$ with respect to $x$ is $u'(x) = -6$ .
Simplify Expression: Now, we apply the chain rule: $h'(x) = f'(u(x)) \cdot u'(x)$ . Substituting the derivatives we found, we get $h'(x) = 5(5 - 6x)^4 \cdot (-6)$ .
Simplify Expression: Now, we apply the chain rule: $h'(x) = f'(u(x)) \cdot u'(x)$ . Substituting the derivatives we found, we get $h'(x) = 5(5 - 6x)^4 \cdot (-6)$ . Simplify the expression by multiplying the constants: $h'(x) = -30(5 - 6x)^4$ .