Question

find the derivative of the function

$y = e^{sqrt{x^{2}-2}}$

using the chain rule for exponential functions.

$\frac{dy}{dx}=$

Explanation:

Step1: Let $u = \sqrt{x^{2}-2}$

$y = e^{u}$

Step2: Find $\frac{dy}{du}$

The derivative of $y = e^{u}$ with respect to $u$ is $\frac{dy}{du}=e^{u}$

Step3: Find $\frac{du}{dx}$

$u=(x^{2}-2)^{\frac{1}{2}}$, using the power - chain rule $\frac{du}{dx}=\frac{1}{2}(x^{2}-2)^{-\frac{1}{2}}\cdot2x=\frac{x}{\sqrt{x^{2}-2}}$

Step4: Apply the chain rule $\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}$

Substitute $\frac{dy}{du}=e^{u}$ and $\frac{du}{dx}=\frac{x}{\sqrt{x^{2}-2}}$ into the chain - rule formula. Since $u = \sqrt{x^{2}-2}$, we have $\frac{dy}{dx}=e^{\sqrt{x^{2}-2}}\cdot\frac{x}{\sqrt{x^{2}-2}}$

Answer:

$\frac{xe^{\sqrt{x^{2}-2}}}{\sqrt{x^{2}-2}}$