Question

hw14 the chain rule (target c4; §3.6) due: thu oct 9, 2025 11:59pm attempt 1 in progress next up: submit assignment details no additional details were added for this assignment. hw14 the chain rule (target c4; §3.6) score: 0/11 answered: 0/11 question 1 use the chain rule to find the derivative of f(x)=5(3x^8 + 10x^6)^12 you do not need to expand out your answer. f(x) = question help: video message instructor submit question

Explanation:

Step1: Identify outer - inner functions

Let $u = 3x^{8}+10x^{6}$, then $y = 5u^{12}$.

Step2: Differentiate outer function

The derivative of $y$ with respect to $u$ is $\frac{dy}{du}=5\times12u^{11}=60u^{11}$.

Step3: Differentiate inner function

The derivative of $u$ with respect to $x$ is $\frac{du}{dx}=(3\times8x^{7}+10\times6x^{5}) = 24x^{7}+60x^{5}$.

Step4: Apply chain rule

By the chain rule $\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}$. Substitute $u = 3x^{8}+10x^{6}$ back in: $f^{\prime}(x)=60(3x^{8}+10x^{6})^{11}(24x^{7}+60x^{5})$.

Answer:

$60(3x^{8}+10x^{6})^{11}(24x^{7}+60x^{5})$