I finished this problem but I wanted some feedback because I don't know if it's entirely correct.
Suppose that functions f and g and their derivatives with respect to x have the following values at x=2.
f(x)=8
g(x)=2
f'(x)=1/3
g'(x)=-3
Find d/dx of f(g(x)) at x=2.
I know the chain rule is this:
f'(g(x)) * g'(x)
So you would write the derivative of the outer function, keeping the inner function (g(x)) as the 'x' value. But in this case, the derivative of the outer function is a number.... I wasn't sure how to approach this problem. How do I figure in the g(x)?
This is the answer (and work) that I got:
f'(g(x)) * g'(x)
1/3 * -3 = -1
Where 1/3 is f'(x) and -3 is g'(x). But I feel like this is incorrect, because I didn't input the value of g(x).
Does anyone know if what I did was right, or how to correct it if it was wrong? I wasn't given an equation for f(x) or g(x) so I assumed we were supposed to rely on the numbers given. Do I need to multiply f'(x) and g(x) to get 2/3?
Thank you
f'(g(x))* g'(x)
g(2) = 2
f'(2) = 1/3
so
f'(g(2)) = 2/3
and
g'(2) = -3
so I get
(2/3)(-3) = -2
Thank you! That makes more sense... because you would say f' of g of x.... so 1/3 of 2 = 2/3. That makes sense!
To find the derivative of f(g(x)) at x=2, you correctly identified that the chain rule should be applied. However, there seems to be a misunderstanding when it comes to evaluating the derivatives at x=2.
Let's go through the problem step by step:
1. Start with the chain rule: d/dx[f(g(x))] = f'(g(x)) * g'(x).
2. Identify the given values:
- f(x) = 8
- g(x) = 2
- f'(x) = 1/3
- g'(x) = -3
3. Evaluate f'(g(x)) * g'(x) at x=2:
- Evaluate f'(g(x)) at x=2:
- f'(g(x)) = f'(2)
- Since f'(x) = 1/3, f'(2) would be 1/3.
- Evaluate g'(x) at x=2:
- g'(2) = -3
- Now, substitute the values into the chain rule formula:
- f'(g(x)) * g'(x) = (1/3) * (-3)
- Simplifying, you get: -1.
Therefore, the derivative of f(g(x)) at x=2 is -1.
You were correct to use the chain rule, and your answer of -1 is indeed correct. The key is to evaluate the derivatives f'(x) and g'(x) at the specific value x=2, rather than multiplying f'(x) and g(x) together. The value of g(x) doesn't need to be included in the derivative calculation; it only specifies the value of x at which we are evaluating the derivative.