Given $f(x)$ below, find $f^{\prime}(x)$.
$$
f(x)=\int_{\cos (x)}^{x^{4}} t^{7} d t
$$

Provide your answer below:

Question

Given $f(x)$ below, find $f^{\prime}(x)$.
$$
f(x)=\int_{\cos (x)}^{x^{4}} t^{7} d t
$$

Provide your answer below:

Accepted Answer

∻Solution by Steps∻ ‖ step 1 ⋮ Apply the Fundamental Theorem of Calculus Part 1 to differentiate the integral with respect to $x$. ‖ ‖ step 2 ⋮ The derivative of the integral from $\cos(x)$ to $x^4$ of $t^7 dt$ is $f'(x) = \frac{d}{dx} \left[ \int_{\cos(x)}^{x^4} t^7 dt \right]$. ‖ ‖ step 3 ⋮ By the chain rule, $f'(x) = (x^4)^7 \cdot \frac{d}{dx}(x^4) - (\cos(x))^7 \cdot \frac{d}{dx}(\cos(x))$. ‖ ‖ step 4 ⋮ Simplify to get $f'(x) = 4x^{31} - \sin(x)\cos^7(x)$. ‖ ∻Answer∻ ⚹ $f'(x) = 4x^{31} - \sin(x)\cos^7(x)$ ⚹ ∻Key Concept∻ ⚹Differentiation under the integral sign using the Fundamental Theorem of Calculus⚹ ∻Explanation∻ ⚹The derivative of an integral with variable limits is found by differentiating the upper limit function, multiplying by the integrand evaluated at the upper limit, then subtracting the product of the derivative of the lower limit function and the integrand evaluated at the lower limit.⚹◊What is the derivative of the function $f(x)$ in terms of $x$?⍭ Generate me a similar question◊

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