Chain rule question: if f is a differentiable function

[Jaimie]

If f is a differntiable function, find the expression for derivatives of the following functions.

a) g(x)= x/ f(x)
b) h(x) [f(x^3)]^2
c) k(x)= sqrt (1 + [f(x)]^2)

First off, I am not even sure what they are asking. I am assuming that they want the derivative for each component of the equation? then to find the derivative for the entire function?

a) really not sure about this one

b) g(x) = x^2 f(x)= x^3
g'(x)= 2x f'(x)= 3x^2
h'(x)= 2(x^3)(3x^2)
h'(x)= (2x^3)(3x^2)
h'(x)= 6x^5

c) g(x)= sqrt (x) h(x)= 1 + x^2
g'(x)= 1/2 x^-1/2 h'(x)= 2x
f'(x)= 1
k'(x)= 1/2 1 + (x^2)^-1/2(2x)
then continue to find equation.


The fact that f(x) is in the equation is throwing me off. Can you explain why you are approaching the problem this way. I am doing my best but we were given this yesterday to solve, but without understanding the question, I am a little at a loss. Thank you so much!

 

[iRaid]

Well for the first one just think of it as any function. f(x) can be x^2, x^3, etc.

So use the quotient rule:
$$f'(x)=\frac{g'(x)h(x)-g(x)h'(x)}{(h(x))^{2}}$$

 

[iandelaney]

For the second one I would use the chain rule ie bring the square down and the 3x^2 out of the inside of the function to obtain (6x^2)(f(x^3)) I think :-)

 

[CAF123]

For the second one, to take into account f is a function of x^3, you should use the chain rule again to differentiate with respect to x^3.

For the third one, you have to apply the chain rule multiple times.