Hi, I was wondering how to do these types of derivatives... I have several to do, but I would like to see only one of them solved so that I can do the others. The course is about several variable calculus...
The problem says:
"If f(x)=x^5, find
f'(5x^4), f(x^2), d/dx(f(x^2)), f'(x^2)"
This is just composition of functions. If I asked you to find f'(x), you'd probably have no problem doing that. Here's a different example:
Find f'(x+3), f(x^5), d/dx(f(x^3)), f'(x^3)
First, let's calculate f'(x).
Now, remember how to do composition of functions?
The second function is even easier, we just compose f with x^5:
f(x^5)=(x^5)^2+2 = x^10+2
Now we're back to derivatives, and some tricky notation. The book is trying to trip you up into thinking that, since d/dx(f(x))=f'(x), then d/dx(f(x^3))=f'(x^3). This is not the case. Remember the chain rule:
d/dx(f(g(x)) = f'(g(x))g'(x)
So, with g(x)=x^3, g'(x)=3x^2 and therefore
Finally, the last one is just there to possibly alert those people who tripped up on the third one that they should reconsider their answer. It's just simple composition, like before: