Need help with a compound function.

Question

Let,

f(10)=5

g(10)=3

f(4)=2

g(4)=10

f′(10)= 1/2

g′(10)=4

f′(4)=−4

g′(4)=−4

 

If h(x) = 2f(x)g(x) and w(x) = f(g(x))

 

(a) Find h(10).

(b) Find h′(10).

(c) Find w(4).

(d) Find w′(4).

Answer 1

Okay, this first one is just basic substitution.

 

find h(10) 

 

h(10) = 2f(10)g(10) = 2 * 5 * 3 = 30

 

 

The second one requires calculating the derivative function for h'(x) using the product rule

 

h'(x) = 2f'(x)g(x) + 2f(x)g'(x)

 

so 

 

h'(10) = 2 * f'(10) * g(10) + 2 * f(10) * g'(10)

h'(10) = 2 * 1/2 * 3 + 2 * 5 * 4 

h'(10) = 3 + 40 = 43

 

and similarly for the other two, but in that case using the chain rule instead of the product rule.

 

w'(x) = f'(g(x))g'(x)

 

 

 

Answer 2

Hi John,

 

For parts a) and c) you just need to plug the values you're given in.

 

a) h(10) = 2*f(10)*g(10)

c) w(4) = f(g(4))

 

For parts b & d, you need to take the derivatives first, using your derivative rules.

 

h(x) = 2f(x)g(x), so it uses the product rules,

h'(x) = 2f'(x)g(x) + 2f(x)g'(x)

h'(10) = 2f'(10)g(10) + 2f(10)g'(10)

 

w(x) = f(g(x)) has nested functions, so use the chain rule,

w'(x) = f'(g(x))*g'(x)

w'(4) = f'(g(4))*g'(4)

 

You should be able to calculate the values from there.