Question

Can I get help with this graph? It involves derivatives and differentiation. I got h'(2) right but need help with h'(1) and h'(3)

Answer 1

Given that `h(x) = f(x) * g(x)`, we know from the product rule that `h'(x) = f'(x)g(x) + f(x)g'(x)`.

Therefore

`h'(1) = f'(1)g(1) + f(1)g'(1)`

The values of `f'(1)` and `g'(1)` will be determined by taking the slope of the graphs at those points.

`f'(1) = (3 - 0)/(2- 0) = 3/2`
`g'(1) = (1 - 0)/(0 - 4) = -1/4`

To determine the values of `f(1) and g(1)`, knowing the equation of the lines would be helpful.

For f(x): `y - 0 = 3/2(x - 0) -> y = 3/2x`
For g(x): `y - 1 = -1/4(x - 0) -> y = -1/4x + 1`

Therefore

`f(1) = 3/2(1)= 3/2`
`g(1) = -1/4(1) + 1 = 3/4`

Piecing all this together:

`h'(1) = 3/2(3/4) + (-1/4)(3/2) = 9/8 - 3/8 = 6/8 = 3/4`

The same process is required for `h(3)`. I'll leave that up to you.

Hopefully this helps!