Find the Derivative - d/dx y=x^(5x)

Problem

d()/d(x)*x(5*x)

Solution

1. Identify the function as a variable base raised to a variable power, which requires logarithmic differentiation.

2. Set the equation to y=x(5*x) and take the natural logarithm of both sides.

ln(y)=ln(x(5*x))

3. Apply the power rule for logarithms to move the exponent in front of the log.

ln(y)=5*x*ln(x)

4. Differentiate both sides with respect to x using the chain rule on the left and the product rule on the right.

1/yd(y)/d(x)=(d(5)*x)/d(x)*ln(x)+5*xd(ln(x))/d(x)

5. Calculate the derivatives of the individual terms.

1/yd(y)/d(x)=5*ln(x)+5*x1/x

6. Simplify the expression on the right side.

1/yd(y)/d(x)=5*ln(x)+5

7. Solve for d(y)/d(x) by multiplying both sides by y

d(y)/d(x)=y*(5*ln(x)+5)

8. Substitute the original expression for y back into the equation.

d(y)/d(x)=x(5*x)*(5*ln(x)+5)

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

d(y)/d(x)=5*x(5*x)*(ln(x)+1)

Final Answer

d(x(5*x))/d(x)=5*x(5*x)*(ln(x)+1)

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