Find the Derivative - d/dx x^(3x)

Problem

d(x(3*x))/d(x)

Solution

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

2. Rewrite the expression using the exponential identity to prepare for the chain rule.

x(3*x)=e(3*x*ln(x))

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

d(e(3*x*ln(x)))/d(x)=e(3*x*ln(x))⋅(d(3)*x*ln(x))/d(x)

4. Apply the product rule to differentiate the exponent 3*x*ln(x)

(d(3)*x*ln(x))/d(x)=3*ln(x)+3*x⋅1/x

5. Simplify the result of the product rule.

3*ln(x)+3

6. Substitute the original expression back in for the exponential term and combine the results.

x(3*x)*(3*ln(x)+3)

7. Factor out the constant to reach the final form.

3*x(3*x)*(ln(x)+1)

Final Answer

d(x(3*x))/d(x)=3*x(3*x)*(ln(x)+1)

Want more problems? Check here!