Calculating Derivative – A function to the power of a function – Exercise 6374


Find the derivative of the following function:


Final Answer

f'(x)=x^{x^2+1}\cdot (2\ln x+1)


We do not have a derivative formula for a function to the power of a function. To work around this, we use a “trick” – we use logarithm rules to get a multiplication of functions instead of a function to the power of a function.


=e^{\ln x^{x^2}}=

=e^{x^2\ln x}

Using Derivative formulas and the multiplication rule and chain rule in Derivative Rules, we get the derivative:

f'(x)=e^{x^2\ln x}\cdot (2x\ln x+x^2\cdot\frac{1}{x})=

One can simplify the derivative:

=e^{\ln x^{x^2}}\cdot (2x\ln x+x)=

By logarithm rules we get:

=x^{x^2}\cdot (2x\ln x+x)=

=x^{x^2+1}\cdot (2\ln x+1)

Have a question? Found a mistake? – Write a comment below!
Was it helpful? You can buy me a cup of coffee here, which will make me very happy and will help me upload more solutions! 

Share with Friends

Leave a Reply