Question

hw14 the chain rule (target c4; §3.6)
score: 2/11 answered: 2/11
question 3
if $f(x)=sin(x^{4})$, find $f(x)$
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Explanation:

Step1: Identify outer - inner functions

Let $u = x^{4}$, then $y=\sin(u)$.

Step2: Differentiate outer function

The derivative of $y = \sin(u)$ with respect to $u$ is $\frac{dy}{du}=\cos(u)$.

Step3: Differentiate inner function

The derivative of $u = x^{4}$ with respect to $x$ is $\frac{du}{dx}=4x^{3}$.

Step4: Apply chain - rule

By the chain - rule $\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}$. Substitute $\frac{dy}{du}=\cos(u)$ and $\frac{du}{dx}=4x^{3}$ back in, and replace $u$ with $x^{4}$. So $\frac{dy}{dx}=\cos(x^{4})\cdot4x^{3}=4x^{3}\cos(x^{4})$.

Answer:

$4x^{3}\cos(x^{4})$