hw12 derivatives of trigonometric functions (target c3; $3.5)\nscore: 4/7 answered: 4/7\nquestion 5\nif…

Explanation:

Step1: Simplify the function

First, recall that $\frac{\tan x}{\sec x}=\sin x$. So $f(x) = 5x^{2}\sin x$.

Step2: Apply the product - rule

The product - rule states that if $y = u\cdot v$, where $u$ and $v$ are functions of $x$, then $y'=u'v + uv'$. Let $u = 5x^{2}$ and $v=\sin x$. Then $u'=10x$ and $v'=\cos x$.

Step3: Calculate the derivative

Using the product - rule $f'(x)=u'v + uv'$, we substitute $u$, $u'$, $v$, and $v'$: $f'(x)=(10x)\sin x+5x^{2}\cos x$.

Answer:

$10x\sin x + 5x^{2}\cos x$