You had the right idea, but you lost you equation, and so lost your way.DollarBill said:Homework Statement
Let
g(5)=-3
g'(5)=6
h(5)=3
h'(5)=-2
Find f'(5) for f(x)=g(h(x))
Find f'(5) for [g(x)]3
The Attempt at a Solution
Find f'(5) for f(x)=g(h(x))
g'(h(x))*h'(x)
g'(3)*-2
But I don't know where to go from there because I'm not given g'(3).
Find f'(5) for [g(x)]3
I was thinking just to use the power rule, but it wasn't right
3g(x)2
3(-3)2
I'm actually not quite sure what you mean...Mark44 said:You had the right idea, but you lost you equation, and so lost your way.
Mark
- Start with the equation for f(x). f(x) = ...
- Find f'(x). This should be an equation. f'(x) = ...
- Evaluate f' at x = 5. f'(5) = ... To do this, you'll need the function values in your problem statement.
Didn't see your reply before. Thanks, but I'm still not sure about the first one.SticksandStones said:For the second one think of it like any ordinary chain rule problem. [g(x)]^3 = h(g(x)) where h(x) = x^3.
Not really. You have an expression for f'(5). Take another look at the information that was given in the problem to make sure you have all of the given information and that you have included it in this thread. If so, and the problem didn't give you a value for g'(3), then you have done everything that you can and -2*g'(3) is the value for f'(5).DollarBill said:I'm actually not quite sure what you mean...
1. Start with equation: f(x)=(g(h(x))
2 Find f'(x): f'(x)=g'(h(x))*h'(x)
3.f'(5)=g'(h(5))*h'(5)
4.f'(5)=g'(3)*-2
And I'm pretty much where I was before...
f'(5) represents the derivative of the function f(x) at the specific value of x = 5. In this case, f(x) is composed of two functions, g(x) and h(x), and we are finding the derivative of this composite function at x = 5.
The derivative of a composite function can be calculated using the chain rule, which states:
f'(x) = g'(h(x)) * h'(x)
In other words, we take the derivative of the outer function (g(x)), evaluated at the inner function (h(x)), multiplied by the derivative of the inner function (h'(x)).
No, the order of the functions cannot be switched. The chain rule relies on the specific order of the functions, with the outer function being evaluated at the inner function. Switching the order would result in a different derivative.
Raising g(x) to the power of 3 is simply a part of the given function and does not affect the process of finding the derivative. It is likely included to demonstrate the calculation of a derivative for a more complex function.
The value of f'(5) represents the slope of the tangent line to the function f(x) at the point x = 5. Since f(x) is composed of the functions g(x) and h(x), the slope at x = 5 is determined by both of these functions and their respective slopes.