Question

given the function $f(x)=x^{-1}cos x$, find $f(x)$ in any form. answer attempt 1 out of 2 $f(x)=$

Explanation:

Step1: Apply product - rule

The product - rule states that if $y = u\cdot v$, then $y^\prime=u^\prime v + uv^\prime$. Here, $u = x^{-1}$ and $v=\cos x$.

Step2: Find $u^\prime$ and $v^\prime$

The derivative of $u = x^{-1}$ using the power - rule $\frac{d}{dx}(x^n)=nx^{n - 1}$ is $u^\prime=-x^{-2}$. The derivative of $v=\cos x$ is $v^\prime=-\sin x$.

Step3: Substitute into product - rule

$f^\prime(x)=u^\prime v+uv^\prime=-x^{-2}\cos x+x^{-1}(-\sin x)=-\frac{\cos x}{x^{2}}-\frac{\sin x}{x}$.

Answer:

$-\frac{\cos x}{x^{2}}-\frac{\sin x}{x}$