Question

homework 3.6 the chain rule
score: 30/100 answered: 3/10
question 4
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if (f(x)=sin(x^{3})), find (f(x))
find (f(1))
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Explanation:

Step1: Identify outer - inner functions

Let $u = x^{3}$, 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 $u = x^{3}$ with respect to $x$ is $\frac{du}{dx}=3x^{2}$.

Step4: Apply chain - rule

By the chain - rule $\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}$. Substituting $\frac{dy}{du}=\cos(u)$ and $\frac{du}{dx}=3x^{2}$, and replacing $u$ with $x^{3}$, we get $f^{\prime}(x)=\cos(x^{3})\cdot3x^{2}=3x^{2}\cos(x^{3})$.

Step5: Evaluate $f^{\prime}(1)$

Substitute $x = 1$ into $f^{\prime}(x)$. Then $f^{\prime}(1)=3\times1^{2}\times\cos(1^{3})=3\cos(1)$.

Answer:

$f^{\prime}(x)=3x^{2}\cos(x^{3})$
$f^{\prime}(1)=3\cos(1)$