Exercise
Find the derivative of the following function:
f(x)=x^{x^2}
Final Answer
Solution
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.
f(x)=x^{x^2}=
=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!