Find the Derivative - d/dx x^(8x)

Problem

d(x^(8*x))/d(x)

Solution

1. Identify the function as a variable base raised to a variable power, which requires logarithmic differentiation or the identity x^(8*x)=e^ln(x^(8*x))

2. Rewrite the expression using the exponential identity a^b=e^(b*ln(a))

x^(8*x)=e^(8*x*ln(x))

3. Apply the chain rule to differentiate the exponential function.

d(e^(8*x*ln(x)))/d(x)=e^(8*x*ln(x))⋅(d(8)*x*ln(x))/d(x)

4. Apply the product rule to differentiate the exponent 8*x*ln(x)

(d(8)*x*ln(x))/d(x)=(d(8)*x)/d(x)⋅ln(x)+8*x⋅d(ln(x))/d(x)

5. Compute the derivatives of the individual terms.

(d(8)*x*ln(x))/d(x)=8*ln(x)+8*x⋅1/x

6. Simplify the resulting expression for the derivative of the exponent.

(d(8)*x*ln(x))/d(x)=8*ln(x)+8

7. Substitute the result back into the chain rule expression and replace e^(8*x*ln(x)) with the original x^(8*x)

d(x^(8*x))/d(x)=x^(8*x)*(8*ln(x)+8)

8. Factor out the common constant to reach the final form.

d(x^(8*x))/d(x)=8*x^(8*x)*(ln(x)+1)

Final Answer

d(x^(8*x))/d(x)=8*x^(8*x)*(ln(x)+1)

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