If h(x) = the square root of 4 + 3f(x)
where f(2) = 4 and f '(2) = 2, find h'(2).
i got the derivative as 1/2(4+3f(x))^-1/2
then i solved for the inside which is (0+3(8))
then multiplied the the two eating and i got 3.... but answer is wrong =(
Oh no, it seems like you made an arithmetic mistake along the way. Let's go through the problem together and find the correct answer, shall we?
We are given that h(x) = √(4 + 3f(x)), where f(2) = 4 and f'(2) = 2. We want to find h'(2).
To find h'(2), let's start by finding the derivative of h(x) with respect to x.
h'(x) = d/dx(√(4 + 3f(x)))
To apply the chain rule, we need to find the derivative of the inner function f(x) with respect to x.
f'(x) = 2
Now, let's substitute these values back into the derivative of h(x):
h'(x) = 1/(2√(4 + 3f(x))) * d/dx(4 + 3f(x))
= 1/(2√(4 + 3f(x))) * 3f'(x)
= 1/(2√(4 + 3f(x))) * 3(2)
= 3/(2√(4 + 3f(x)))
Now let's evaluate h'(2):
h'(2) = 3/(2√(4 + 3f(2)))
= 3/(2√(4 + 3(4)))
= 3/(2√(4 + 12))
= 3/(2√16)
= 3/(2 * 4)
= 3/8
So the correct answer is h'(2) = 3/8.
Don't worry, mistakes happen! Just remember to double-check your calculations. Keep up the good work!
To find h'(2), we need to take the derivative of h(x) with respect to x and evaluate it at x = 2.
Given h(x) = √(4 + 3f(x)), we can find h'(x) by applying the chain rule. The chain rule states that if we have a composite function f(g(x)), the derivative of f with respect to x is given by f'(g(x)) * g'(x).
Let's take the derivative step-by-step:
1. First, let's find the derivative of f(x). Given f(2) = 4 and f '(2) = 2, we know that f'(x) = 2.
2. Now, let's find the derivative of h(x):
h'(x) = d/dx [√(4 + 3f(x))]
Using the chain rule, we have:
h'(x) = (1/2) * (4 + 3f(x))^(-1/2) * d/dx[4 + 3f(x)]
3. We want to evaluate h'(x) at x = 2, so let's substitute x = 2 into the expression:
h'(2) = (1/2) * (4 + 3f(2))^(-1/2) * d/dx[4 + 3f(2)]
Substituting f(2) = 4 and f '(2) = 2, we have:
h'(2) = (1/2) * (4 + 3(4))^(-1/2) * d/dx[4 + 3(4)]
Simplifying further:
h'(2) = (1/2) * (16 + 12)^(-1/2) * d/dx[16 + 12]
h'(2) = (1/2) * (28)^(-1/2) * 0
Since d/dx[16 + 12] = 0, the final result is:
h'(2) = 0
Therefore, the correct answer is h'(2) = 0, not 3.
To find the derivative of h(x), first apply the chain rule. The chain rule states that if you have a composition of functions, such as h(x) = g(f(x)), then the derivative of h(x) with respect to x is equal to the derivative of g(f(x)) with respect to f(x), multiplied by the derivative of f(x) with respect to x.
In this case, h(x) = √(4 + 3f(x)). To differentiate h(x) with respect to x, we need to find both the derivative of the outer function (√( )) and the derivative of the inner function (4 + 3f(x)).
Now let's find the derivative of h(x) step by step:
Step 1: Find the derivative of the outer function:
The derivative of √u with respect to u is 1 / (2√u).
So, applying the derivative of the outer function to h(x), we get:
h'(x) = (1 / (2√(4 + 3f(x)))) * (d / dx) (4 + 3f(x))
Step 2: Find the derivative of the inner function:
The derivative of 4 + 3f(x) with respect to x is equal to the derivative of 4 with respect to x plus the derivative of 3f(x) with respect to x. Since f(x) is a function of x, we use the product rule to differentiate 3f(x).
The derivative of 4 with respect to x is 0 because it's a constant.
The derivative of 3f(x) with respect to x can be found using the chain rule. The chain rule states that if you have a function of the form f(u) = c(g(u)), then the derivative of f(u) with respect to x is equal to c multiplied by the derivative of g(u) with respect to x.
In this case, let u = f(x) and g(u) = 3u. So the derivative of 3f(x) with respect to x is:
(d / dx) (3f(x)) = 3(d / dx) (f(x)) = 3f'(x)
Now, substituting back into Step 1, we have:
h'(x) = (1 / (2√(4 + 3f(x)))) * (d / dx) (4 + 3f(x))
h'(x) = (1 / (2√(4 + 3f(x)))) * (0 + 3f'(x))
Step 3: Evaluate h'(2):
To find h'(2), we substitute x = 2 into the expression we derived above. Also, use the given information that f(2) = 4 and f'(2) = 2.
h'(2) = (1 / (2√(4 + 3f(2)))) * (0 + 3f'(2))
h'(2) = (1 / (2√(4 + 3 * 4))) * (0 + 3 * 2)
Now, simplify and calculate:
h'(2) = (1 / (2√(4 + 12))) * (0 + 6)
h'(2) = (1 / (2√16)) * 6
h'(2) = (1 / (2 * 4)) * 6
h'(2) = (1 / 8) * 6
h'(2) = 6 / 8
h'(2) = 3 / 4
So, the correct answer for h'(2) is 3/4.
h = √(4+3f)
h' = 3/(2√(4+3f)) f'
at x=2,
dh/dx = 3/(2√(4+12)) (2) = 6/8 = 3/4