Trigonometric derivative?

Question

Hello everyone how would I solve the following derivative.

$f(x)=5x^3\tan(x)+\cot(2x)$

I know the derivative of $\tan(x)$ is $\sec^2(x)$

So would I do

$15x^2\sec^2(x)-\csc(2x)$

As my derivative.

Answer 1

Notice that you have a product of two functions in the first term: $$ 5x^3\tan(x). $$ So you need to use the product rule. You get (for the first term only) $$ \frac{d}{dx} 5x^3\tan(x) = \left[\frac{d}{dx}5x^3\right]\tan(x) + 5x^3\frac{d}{dx}\tan(x). $$

Also for the second term $$ \cot(2x) $$ you need to multiply by the derivative of the "inner function" $2x$ (using the Chain Rule here): $$ \frac{d}{dx} \cot(2x) = -\csc^{\color{red} 2}(2x)\frac{d}{dx}(2x). $$