Given f and g are differentiable functions and

f(a)=-4, g(a)=c, g(c)=10, f(c)=15
f'(a)=8, g'(a)=b, g'(c)=5, f'(c)=6

If h(x)=f(g(x)). find h'(a)?

i'm not really sure how to get the answer. g'(a)=b, f'(b)=? How do i go about doing this. My reasoning must be wrong.

h'(x)= f'(g(x))*g'(x)
if x=a
h'(a)=f'(g(a))*g'(a)
=f'(c) * b = 6b

check my thinking.

To find h'(a), the derivative of h(x) with respect to x evaluated at a, you can use the chain rule.

The chain rule states that if you have a composite function such as h(x) = f(g(x)), then the derivative of h(x) with respect to x is given by h'(x) = f'(g(x)) * g'(x).

Applying the chain rule to h(x) = f(g(x)), you get:
h'(x) = f'(g(x)) * g'(x)

Now, let's evaluate this equation at x = a to find h'(a):
h'(a) = f'(g(a)) * g'(a)

From the given information, we know that:
g(a) = c
f(a) = -4
f(c) = 15

Therefore, we can rewrite the equation as follows:
h'(a) = f'(c) * g'(a)

From the given information, we also know that:
f'(c) = 6
g'(a) = b

Substituting these values into the equation, we get:
h'(a) = 6 * b

So, the derivative of h(x) with respect to x evaluated at a, h'(a), is 6b.

Therefore, your thinking is correct. h'(a) is indeed equal to 6b.