h(x)=(t+1)^1/2
k(-2)=2
k(2)=3
k'(-2)=9
k'(2)=10
find derivative h(k(1))
h(k(x)) at x=2
find derivative h(k(2))
I think you have garbled the problem.
How can h(x) = √(t+1) ?
Fix that, and then you can use the chain rule.
To find the derivative of h(k(1)), we'll need to use the chain rule. Here's how you can do it step by step:
1. Start by finding the derivative of h(x) with respect to x. The function h(x)=(t+1)^1/2 can be rewritten as h(x)=(x+1)^1/2. To find its derivative, we apply the power rule for differentiation, which states that d/dx(x^n) = n*x^(n-1).
So, applying the power rule, the derivative of h(x) = (x+1)^1/2 is h'(x) = (1/2)*(x+1)^(-1/2).
2. Next, we'll find the derivative of k(x) with respect to x. We are given that k(-2) = 2 and k(2) = 3. These values don't directly help us find the derivative of k(x), so we'll use them later.
3. Now, let's find k'(x). We are given that k'(-2) = 9 and k'(2) = 10. Since these are the values of the derivative of k(x) at x = -2 and x = 2, respectively, we can directly use them.
So, k'(x) = 9 when x = -2 and k'(x) = 10 when x = 2.
4. Now, we can find the derivative of h(k(x)) with respect to x using the chain rule. The chain rule states that if we have a composite function F(g(x)), where F(u) and g(x) are both differentiable functions, then the derivative of F(g(x)) with respect to x is equal to the derivative of F(u) with respect to u, multiplied by the derivative of g(x) with respect to x.
Applying the chain rule, the derivative of h(k(x)) with respect to x is h'(k(x)) * k'(x).
5. In the case of h(k(1)), substitute x = 1 into the functions k(x) and h'(k(x)). We have k(1) = 3 and h'(k(1)) = h'(3). Since we don't know the direct value of h'(3), we'll need to find it in step 6.
6. To find h'(3), we substitute k(1) = 3 into k'(x), which gives us k'(1) = 10. Then substitute k'(1) = 10 into h'(x), which gives us h'(3) = (1/2)*(3+1)^(-1/2) = (1/2)*(4)^(-1/2) = 1/4.
7. Finally, substitute h'(3) = 1/4 back into our derivative expression in step 4, and we have h'(k(1)) = h'(3) * k'(1) = (1/4) * 10 = 10/4 = 5/2.
So, the derivative of h(k(1)) is 5/2.
Now, let's move on to finding h(k(x)) at x = 2.
1. Start with the function h(x) = (x+1)^1/2.
2. Substitute k(x) = 2 into h(x), which gives us h(k(x)) = h(2) = (2+1)^1/2 = 3^1/2 = √3.
So, h(k(x)) at x = 2 is √3.
I hope this explanation helps you understand how to find the derivative h(k(1)) and evaluate h(k(x)) at x = 2 using the given information.