Cheyanne Leigh

2021-02-21

In this question, the function f is differentiable, and f'(x) = g(x). We don't know exactly what f(x) or g(x) are, so your answers will have f(x) and g(x) in them.
Compute the derivatives of the following function.
${e}^{\mathrm{sin}\left(f\left(x\right)\right)}$

### Answer & Explanation

Raheem Donnelly

Use chain rule to compute the derivatives.
$\frac{d}{dx}\left({e}^{\mathrm{sin}\left(f\left(x\right)\right)}\right)=\frac{d}{d\left(\mathrm{sin}\left(f\left(x\right)\right)\right)}\left({e}^{\mathrm{sin}\left(f\left(x\right)\right)}\right)×\frac{d}{d\left(f\left(x\right)\right)}\left(\mathrm{sin}\left(f\left(x\right)\right)\right)×\frac{d}{dx}f\left(x\right)$
$={e}^{\mathrm{sin}\left(f\left(x\right)\right)}×\mathrm{cos}\left(f\left(x\right)\right)×{f}^{\prime }\left(x\right)$
$={e}^{\mathrm{sin}\left(f\left(x\right)\right)}×\mathrm{cos}\left(f\left(x\right)\right)×g\left(x\right)$
$={e}^{\mathrm{sin}\left(f\left(x\right)\right)}g\left(x\right)\mathrm{cos}\left(f\left(x\right)\right)$

Jeffrey Jordon

Answer is given below (on video)

Jeffrey Jordon