Find the Derivative - d/dx ( natural log of x)^5

Problem

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

Solution

1. Identify the outer and inner functions to apply the chain rule. The outer function is u5 and the inner function is u=ln(x)

2. Apply the power rule to the outer function by bringing the exponent to the front and subtracting one from the exponent.

d()/d(x)*(ln(x))5=5*(ln(x))4⋅d(ln(x))/d(x)

3. Differentiate the inner function ln(x) with respect to x

d(ln(x))/d(x)=1/x

4. Multiply the results together to find the final derivative.

d()/d(x)*(ln(x))5=5*(ln(x))4⋅1/x

5. Simplify the expression into a single fraction.

d()/d(x)*(ln(x))5=(5*(ln(x))4)/x

Final Answer

d()/d(x)*(ln(x))5=(5*(ln(x))4)/x

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