# How do you differentiate y=7^(x^2)?

Feb 5, 2017

$y = {7}^{{x}^{2}} \implies y ' = \left(2 \ln \left(7\right) x\right) {7}^{{x}^{2}}$

#### Explanation:

Since $\forall r \in \mathbb{R} , {x}^{r} = {e}^{r \ln \left(x\right)}$

It is true that

${7}^{{x}^{2}} = {e}^{{x}^{2} \ln \left(7\right)}$

Then by the chain rule $\left(f \left(g \left(x\right)\right)\right) ' = f ' \left(g \left(x\right)\right) g ' \left(x\right)$

$\implies \left({e}^{{x}^{2} \ln \left(7\right)}\right) ' = {e}^{{x}^{2} \ln \left(7\right)} 2 \ln \left(7\right) x = {7}^{{x}^{2}} \left(2 \ln \left(7\right) x\right)$