Question

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^{\sin(f(x))}

Derivatives
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.
$$\displaystyle{e}^{{{\sin{{\left({f{{\left({x}\right)}}}\right)}}}}}$$

$$\displaystyle{\frac{{{d}}}{{{\left.{d}{x}\right.}}}}{\left({e}^{{{\sin{{\left({f{{\left({x}\right)}}}\right)}}}}}\right)}={\frac{{{d}}}{{{d}{\left({\sin{{\left({f{{\left({x}\right)}}}\right)}}}\right)}}}}{\left({e}^{{{\sin{{\left({f{{\left({x}\right)}}}\right)}}}}}\right)}\times{\frac{{{d}}}{{{d}{\left({f{{\left({x}\right)}}}\right)}}}}{\left({\sin{{\left({f{{\left({x}\right)}}}\right)}}}\right)}\times{\frac{{{d}}}{{{\left.{d}{x}\right.}}}}{f{{\left({x}\right)}}}$$
$$\displaystyle={e}^{{{\sin{{\left({f{{\left({x}\right)}}}\right)}}}}}\times{\cos{{\left({f{{\left({x}\right)}}}\right)}}}\times{f}'{\left({x}\right)}$$
$$\displaystyle={e}^{{{\sin{{\left({f{{\left({x}\right)}}}\right)}}}}}\times{\cos{{\left({f{{\left({x}\right)}}}\right)}}}\times{g{{\left({x}\right)}}}$$
$$\displaystyle={e}^{{{\sin{{\left({f{{\left({x}\right)}}}\right)}}}}}{g{{\left({x}\right)}}}{\cos{{\left({f{{\left({x}\right)}}}\right)}}}$$