Find the Derivative - d/dx y=4^(e^(x^2))

Problem

d()/d(x)*4^(e^(x^2))

Solution

1. Identify the outer function as an exponential function with base 4 which follows the rule d()/d(x)*a^u=a^u*ln(a)d(u)/d(x)

2. Apply the chain rule to the base 4 exponential, treating e^(x^2) as the inner function u

d(4^(e^(x^2)))/d(x)=4^(e^(x^2))*ln(4)d(e^(x^2))/d(x)

3. Differentiate the inner exponential function e^(x^2) using the rule d()/d(x)*e^v=e^vd(v)/d(x)

d(e^(x^2))/d(x)=e^(x^2)d(x^2)/d(x)

4. Apply the power rule to find the derivative of the innermost function x^2

d(x^2)/d(x)=2*x

5. Substitute the results back into the chain rule expression.

d(4^(e^(x^2)))/d(x)=4^(e^(x^2))*ln(4)*e^(x^2)*(2*x)

6. Rearrange the terms into a standard simplified form.

d(4^(e^(x^2)))/d(x)=2*x*ln(4)*e^(x^2)*4^(e^(x^2))

Final Answer

d()/d(x)*4^(e^(x^2))=2*x*ln(4)*e^(x^2)*4^(e^(x^2))

Want more problems? Check here!