How to Find the Derivatives of $\tan^2(x^4)$ and $\sec^3(x^5)$?

Question

I am to find the derivative of f(x) and g(x):

So far, I know the following:

• The derivative of tan(x) = sec(x)^2
• The derivative of sec(x) = sec(x)tan(x)

So, I have tried the following steps to get the derivative of f(x):

1) Convert to sec(x) using the chain rule --> (sec(x^4)*4x^3)^2

2) This equals ((4x^3)(sec(x^4))^2

For g(x), however, I am not quite sure where to begin.

As well, I do not believe that ((4x^3)(sec(x^4))^2 = f'(x).

Can anyone please show me where I went wrong trying to find the derivative of f(x), as well as point me in the right direction in finding g(x)?

Answer 1

Hint. By the chain rule, you rather have $$ (\tan^2(x^4))'=2\times 4x^3\times \sec^2(x^4)\times\tan(x^4) $$ and $$ (\sec^3(x^5))'=3\times 5x^4\times \sec(x^5)\times\tan (x^5)\times\sec^2(x^5) $$

Answer 2

Use the Chain rule three times:

$\frac{d}{dx}($tan$^2(x^4))=(2)$tan$(x^4)\cdot\sec^2(x^4)\cdot 4x^3=\frac{8x^3 sin(x^4)}{cos^3(x^4)}$. Try for the second one.