Let h(x) be a function which satisfies h'(4) = 3. If k(x) = h(x2), compute k'(2).
To compute k'(2), we need to first find the derivative of the function k(x). Given that k(x) = h(x^2), we can use the chain rule to differentiate it.
The chain rule states that if we have a function g(x) = f(h(x)), then the derivative of g(x) with respect to x is given by g'(x) = f'(h(x)) * h'(x).
In our case, k(x) = h(x^2). So using the chain rule, we have:
k'(x) = h'(x^2) * (x^2)'
To find (x^2)', we differentiate x^2 with respect to x, which results in 2x.
Now we need to find h'(x^2). We are given that h'(4) = 3. Note that x^2 = 4 when x = 2.
Using the chain rule again, we can find h'(x^2) by substituting x^2 = 4 into the derivative h'(x) and evaluating it at x = 2.
Therefore, h'(x^2) evaluated at x = 2 is the same as h'(4), which is 3.
Substituting all the values into our previous equation:
k'(x) = h'(x^2) * (x^2)' = 3 * 2x = 6x
Now we can find k'(2) by substituting x = 2 into the expression we just derived:
k'(2) = 6(2) = 12
Therefore, k'(2) = 12.