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

Answers (1)

2021-02-23
Use chain rule to compute the derivatives.
\(\displaystyle{\frac{{{d}}}{{{\left.{d}{x}\right.}}}}{\left({e}^{{{\sin{{\left({f{{\left({x}\right)}}}\right)}}}}}\right)}={\frac{{{d}}}{{{d}{\left({\sin{{\left({f{{\left({x}\right)}}}\right)}}}\right)}}}}{\left({e}^{{{\sin{{\left({f{{\left({x}\right)}}}\right)}}}}}\right)}\times{\frac{{{d}}}{{{d}{\left({f{{\left({x}\right)}}}\right)}}}}{\left({\sin{{\left({f{{\left({x}\right)}}}\right)}}}\right)}\times{\frac{{{d}}}{{{\left.{d}{x}\right.}}}}{f{{\left({x}\right)}}}\)
\(\displaystyle={e}^{{{\sin{{\left({f{{\left({x}\right)}}}\right)}}}}}\times{\cos{{\left({f{{\left({x}\right)}}}\right)}}}\times{f}'{\left({x}\right)}\)
\(\displaystyle={e}^{{{\sin{{\left({f{{\left({x}\right)}}}\right)}}}}}\times{\cos{{\left({f{{\left({x}\right)}}}\right)}}}\times{g{{\left({x}\right)}}}\)
\(\displaystyle={e}^{{{\sin{{\left({f{{\left({x}\right)}}}\right)}}}}}{g{{\left({x}\right)}}}{\cos{{\left({f{{\left({x}\right)}}}\right)}}}\)
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