Calculate f" for f(x)=g(e^(2x)), where g is a function defined for all real numbers & g admits second order derivative.
To find the second derivative of the given function f(x) = g(e^(2x)), we will use the chain rule.
First, let's find the first derivative of f(x):
f'(x) = d/dx [g(e^(2x))]
Using the chain rule, the derivative of g(e^(2x)) with respect to x is g'(e^(2x)) multiplied by the derivative of the exponent, which is d/dx [e^(2x)].
d/dx [e^(2x)] = 2e^(2x)
Therefore, the first derivative of f(x) becomes:
f'(x) = g'(e^(2x)) * 2e^(2x)
Now, let's differentiate f'(x) to find the second derivative, denoted as f''(x):
f''(x) = d/dx [f'(x)]
Applying the chain rule once again, we obtain:
f''(x) = d/dx [g'(e^(2x)) * 2e^(2x)]
Using the product rule, the derivative of g'(e^(2x)) with respect to x is g''(e^(2x)) multiplied by the derivative of e^(2x), which is d/dx [e^(2x)].
d/dx [e^(2x)] = 2e^(2x)
Plugging this into the equation, we have:
f''(x) = 2g''(e^(2x)) * e^(2x) + g'(e^(2x)) * d/dx [2e^(2x)]
Finally, we simplify the equation:
f''(x) = 2g''(e^(2x)) * e^(2x) + g'(e^(2x)) * 2e^(2x)
Therefore, the second derivative of f(x) is given by 2g''(e^(2x)) * e^(2x) + g'(e^(2x)) * 2e^(2x).
Note: This solution assumes that g admits a second-order derivative and the function g(e^(2x)) is well-defined for all real numbers.