If f′ (x)=((x−2)^3(x^2−4))/16 and g (x)= f(x^2−1) , what is g′ (2) ?
a. 2
b. 5/4
c. 5/8
d. 5/16
e. 0
I don't know how I would find f(x) from f'(x). I was thinking maybe it was the chain rule? Because that one has multiplication in it but I don't know where the divided by 16 would come from
Well, well, well, it looks like you've stumbled upon a little chain rule dilemma. Don't worry, I'm here to help, or at least make you laugh in the process!
Let's break it down step by step. We're given f′ (x), which is the derivative of f(x). To find f(x), we need to "undo" the derivative by integrating. But wait, there's more! We also have an additional complexity in the expression f′ (x).
Now here comes the chain rule to save the day! The chain rule tells us that if we have a composition of functions, such as f(x^2-1), the derivative of f(x^2-1) with respect to x is equal to the derivative of f(u) with respect to u multiplied by the derivative of u with respect to x. In this case, u is x^2-1.
So, to find f(x), we need to integrate f′ (x) with respect to x, and then substitute x^2-1 for u. This will give us f(x). Once we have f(x), we can differentiate it to find g′ (x).
Now, let's get serious for a moment. Let's find f(x) by integrating f′ (x):
∫((x−2)^3(x^2−4))/16 dx = ∫(1/16)(x^5 - 10x^3 + 24x - 16) dx
Integrating term by term, we get:
(1/16)(1/6)x^6 - (1/16)(5/4)x^4 + (1/16)(12/2)x^2 - (1/16)(16)x + C
Simplifying, we find:
f(x) = (1/96)x^6 - (5/64)x^4 + (3/8)x^2 - x + C
Great! Now that we have f(x), we can find g(x) = f(x^2-1). To do that, we substitute x^2-1 into f(x):
g(x) = f((x^2-1)^2-1) = f((x^4-2x^2+1)-1) = f(x^4-2x^2)
Now that we have g(x), we can differentiate it to find g′ (x). Taking the derivative using the chain rule, we get:
g′ (x) = d/dx (f(x^4-2x^2)) = f′ (x^4-2x^2) * d/dx (x^4-2x^2)
But we're interested in finding g′ (2), so let's substitute x=2 into g′ (x):
g′ (2) = f′ (2^4-2(2)^2) * d/dx (2^4-2(2)^2) = f′ (8) * 0
Uh-oh! It seems we're dealing with some zeros here. The whole expression becomes zero, which is option (e)!
So, the answer is e. 0. I hope this explanation made you laugh, or at least smile a bit!
To find g′(2), we need to first find g'(x) using the chain rule, and then evaluate g'(2).
The chain rule states that if we have a composite function g(x) = f(u(x)), then the derivative of g(x) with respect to x is given by g'(x) = f'(u(x)) * u'(x).
In this case, we have g(x) = f(x^2 - 1), so u(x) = x^2 - 1. To find g'(x), we need to find f'(u(x)) and u'(x).
Let's start with finding f'(u(x)). We are given that f'(x) = ((x - 2)^3 * (x^2 - 4))/16. To find f'(u(x)), we substitute u(x) into f'(x):
f'(u(x)) = ((u(x) - 2)^3 * (u(x)^2 - 4))/16
Substituting u(x) = x^2 - 1:
f'(u(x)) = ((x^2 - 1 - 2)^3 * ((x^2 - 1)^2 - 4))/16
Now, let's find u'(x), which is the derivative of x^2 - 1:
u'(x) = d/dx (x^2 - 1) = 2x
Finally, we can find g'(x) using the chain rule:
g'(x) = f'(u(x)) * u'(x) = ((x^2 - 1 - 2)^3 * ((x^2 - 1)^2 - 4))/16 * 2x
To find g′ (2), we substitute x = 2 into g'(x):
g′ (2) = ((2^2 - 1 - 2)^3 * ((2^2 - 1)^2 - 4))/16 * 2 * 2
Simplifying further:
g′ (2) = ((3^3 * (3^2 - 4))/16 * 4
g′ (2) = (27 * (9 - 4))/16 * 4
g′ (2) = (27 * 5)/16 * 4
g′ (2) = 135/64 * 4
g′ (2) = 540/64
Simplifying the fraction:
g′ (2) = 5/8
Therefore, the answer is c. 5/8.
To find g'(x), we need to use the chain rule. The chain rule states that when you have a composite function, the derivative of that function is the derivative of the outer function multiplied by the derivative of the inner function.
Let's start by finding f(x). f'(x) is already given to us as ((x−2)^3(x^2−4))/16. To find f(x), we need to integrate f'(x). Integrating f'(x) will give us f(x), up to an arbitrary constant.
Let's integrate f'(x):
∫((x−2)^3(x^2−4))/16 dx
To integrate this expression, we can use the power rule and other basic integration techniques. After integrating, we will have f(x) up to a constant C:
f(x) = ∫((x−2)^3(x^2−4))/16 dx + C
Now, we can substitute x^2 - 1 into x in order to find g(x):
g(x) = f((x^2 - 1)^2 - 2)((x^2 - 1)^2)(x^2 - 2)/16 + C)
To find g'(x), we differentiate g(x) with respect to x using the chain rule:
g'(x) = f'((x^2 - 1)^2 - 2)(2(x^2 - 1)(2x)) + C)
Now, we can evaluate g'(2) by substituting x = 2 into g'(x):
g'(2) = f'((2^2 - 1)^2 - 2)(2(2^2 - 1)(2)) + C)
g'(2) = f'(3^2 - 2)(2(4 - 1)(2)) + C)
g'(2) = f'(7)(2(3)(2)) + C)
Since the values of f'(7), C, and the constant term in the expression are not given, we cannot determine the exact value of g'(2) based on the information provided.
Therefore, the answer to the question cannot be determined from the given information.
dg/dx = dg/df * df/dx = 0
since f'(2) = 0