find f' for each of the following where g and k are differentiable functions.
F= g{sq.rt. of k(x^2)}
To find the derivative of the function F = g(sqrt(k(x^2))), we can apply the chain rule.
First, let's consider the function inside the square root: k(x^2). The derivative of this function with respect to x can be found using the power rule and the chain rule:
d/dx [k(x^2)] = 2kx
Next, let's consider the function inside the composition: sqrt(k(x^2)). The derivative of this function with respect to x can be found using the chain rule:
d/dx[sqrt(k(x^2))] = 1 / (2 * sqrt(k(x^2))) * d/dx [k(x^2)]
Combining these two derivatives, we can find the derivative of F with respect to x:
d/dx [F] = g'(sqrt(k(x^2))) * d/dx [sqrt(k(x^2))]
Using the chain rule again, we have:
d/dx [F] = g'(sqrt(k(x^2))) * 1 / (2 * sqrt(k(x^2))) * d/dx [k(x^2)]
Substituting the derivative of k(x^2) we found earlier:
d/dx [F] = g'(sqrt(k(x^2))) * 1 / (2 * sqrt(k(x^2))) * 2kx
Simplifying further:
d/dx [F] = g'(sqrt(k(x^2))) * kx / sqrt(k(x^2))
Therefore, the derivative of F = g(sqrt(k(x^2))) with respect to x is:
F' = g'(sqrt(k(x^2))) * kx / sqrt(k(x^2))