Question

question given the function f(x)=cos(4√x), find f(x). answer attempt 1 out of 2 f(x)=

Explanation:

Step1: Identify outer - inner functions

Let $u = 4\sqrt{x}=4x^{\frac{1}{2}}$, then $y=\cos(u)$.

Step2: Differentiate outer function

The derivative of $y = \cos(u)$ with respect to $u$ is $y^\prime_{u}=-\sin(u)$.

Step3: Differentiate inner function

The derivative of $u = 4x^{\frac{1}{2}}$ with respect to $x$ is $u^\prime_{x}=4\times\frac{1}{2}x^{-\frac{1}{2}} = \frac{2}{\sqrt{x}}$.

Step4: Apply chain - rule

By the chain - rule $\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}$, we substitute $y^\prime_{u}$ and $u^\prime_{x}$:
$f^\prime(x)=-\sin(4\sqrt{x})\cdot\frac{2}{\sqrt{x}}=-\frac{2\sin(4\sqrt{x})}{\sqrt{x}}$.

Answer:

$-\frac{2\sin(4\sqrt{x})}{\sqrt{x}}$