Question

How do I find the second derivative of y=sec^3(pi(x))? I've found the first derivative, y'=3pi(sec^3(pi(x))(tan(pi(x)), but don't know what rules to use to find the derivative of y'. Thanks

Answer 1

For composite functions like this one, you need the chain rule. It might help to rewrite some of the given notation to see what the next step would be.

The chain rule:
\[\frac{d}{dx}f(g(x))=f'(g(x))\times \frac{d}{dx}g(x)\]
Given function
\[y=\sec^3(\pi x)=\left(\sec(\pi x)\right)^3\]
Here's what you can do to make it easier to see what function goes where if you want to use the formula above. Let \(f(x)=x^3\) and \(g(x)=\sec(\pi x)\), then
\[y=f(g(x))=f(\sec(\pi x))=(\sec(\pi x))^3\]
Differentiating by the chain rule gives
\[\frac{d}{dx}(\sec(\pi x))^3=3(\sec(\pi x))^2\times\frac{d}{dx}\sec(\pi x)\]
Got it so far?

Answer 2

yea, so onwards you would keep 3secxpix^2, and use chain rule again to find d/dx of secpix, which is
\[\pi*\tan(\pi x)*\sec(\pi x)\]yes?

Answer 3

Yeah that's right. You might want to use a more systematic approach by applying the same reasoning.
\[3(\sec(\pi x))^2\times\color{red}{\frac{d}{dx}\sec(\pi x)}=3\sec^2(\pi x)\color{red}{\sec(\pi x)\tan(\pi x)\times\frac{d}{dx}\pi x}\]
and the next step would be
\[3\sec^2(\pi x)\sec(\pi x)\tan(\pi x)\times\color{blue}{\frac{d}{dx}\pi x}=3\sec^2(\pi x)\sec(\pi x)\tan(\pi x)\times\color{blue}{\pi}\]
and simplify to the answer,
\[\frac{d}{dx}\sec^3(\pi x)=3\pi\sec^3(\pi x)\tan(\pi x)\]

Answer 4

Thank you!

Answer 5

yw