Derivative and chain rule questions

Question

1) If g(x)=2ln(x+1) and f is a differentiable function of x what is equivalent to the derivative of f(g(x)) with respect to x?

2) For which of the following functions is the chain rule an appropriate method to find the derivative with respect to x?

a) y=cos(sqrt(x)+1)

b) y=(2^x)(sinx)

c) y= 20/((40x^2)-1)

3) Let f be a differentiable function. If h(x)=(2+f(sinx))^3, what gives the correct process for finding h'(x)?

a) h'(x)=3(2+f(sinx))^2

b) h'(x)=3(2+f(sinx))^2 * f'(sinx)

c) h'(x)=3(2+f(sinx))^2 * f'(cosx)

d) h'(x)=3(2+f(sinx))^2 * f'(sinx)*cosx

Answer 1

1) [f(g(x))] ' = 2/(x+1)·f ' (2ln(x+1))

2) Choose the answer that has a composite function (a function of a function, eg sqrtcosx, e^(5x-1)).

3) h is a function of a function of a function. So differentiate from the inside out:

Deriv of innermost = cosx , Deriv of middle = f ' (sinx) , Deriv of outermost = 3·(2 + f(sinx))^2).

The answer is the product of those 3 derivatives.