find f'(2) given g(2)=3, h(2)=-1, h'(2)=4, and g'(2)=-2
f(x)=g(x)h(x)
I'm just not understanding how to start this at all and what it wants me to derive when I don't know the function
You don't actually have to know the whole function to get its value at one point, if you have other useful information.
You can - and should! - look up the background to "Leibniz's law", but it says that
f(x)=g(x)h(x)
implies
f'(x)=g'(x)h(x) + g(x)h'(x)
Now, this is true for all x, but you only need the value for one x, and I think you can handle it from there.
f'
thanks so much!
You're very welcome, Bri. And do look up _why_ that is true sometime. It's quite neat.
To find f'(2), we need to use the product rule, which states that if f(x) = g(x)h(x), then f'(x) = g'(x)h(x) + g(x)h'(x).
Given g(2) = 3, h(2) = -1, g'(2) = -2, and h'(2) = 4, we can substitute these values into the product rule formula:
f'(2) = g'(2)h(2) + g(2)h'(2)
Substituting the given values:
f'(2) = (-2)(-1) + (3)(4)
Now, we can simplify the expression:
f'(2) = 2 + 12
f'(2) = 14
Therefore, f'(2) = 14.