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

Problem

d()/d(x)*4(e(x2))

Solution

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

2. Apply the chain rule to the base 4 exponential, treating e(x2) as the inner function u

d(4(e(x2)))/d(x)=4(e(x2))*ln(4)d(e(x2))/d(x)

3. Differentiate the inner exponential function e(x2) using the rule d()/d(x)*ev=evd(v)/d(x)

d(e(x2))/d(x)=e(x2)d(x2)/d(x)

4. Apply the power rule to find the derivative of the innermost function x2

d(x2)/d(x)=2*x

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

d(4(e(x2)))/d(x)=4(e(x2))*ln(4)*e(x2)*(2*x)

6. Rearrange the terms into a standard simplified form.

d(4(e(x2)))/d(x)=2*x*ln(4)*e(x2)*4(e(x2))

Final Answer

d()/d(x)*4(e(x2))=2*x*ln(4)*e(x2)*4(e(x2))

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