Question

if (f(x)=sin(x^{4})), find (f(x)).
find (f(5)).
question help: video

Explanation:

Step1: Apply chain - rule

Let $u = x^{4}$, then $f(x)=\sin(u)$. The chain - rule states that $f^\prime(x)=\frac{df}{du}\cdot\frac{du}{dx}$. The derivative of $\sin(u)$ with respect to $u$ is $\cos(u)$, and the derivative of $u = x^{4}$ with respect to $x$ is $4x^{3}$. So $f^\prime(x)=\cos(x^{4})\cdot4x^{3}=4x^{3}\cos(x^{4})$.

Step2: Evaluate $f^\prime(5)$

Substitute $x = 5$ into $f^\prime(x)$. We have $f^\prime(5)=4\times5^{3}\cos(5^{4})$. Calculate $5^{3}=125$, so $f^\prime(5)=4\times125\cos(625)=500\cos(625)$.

Answer:

$f^\prime(x)=4x^{3}\cos(x^{4})$
$f^\prime(5)=500\cos(625)$