If-f-x-x-2-g-x-sin-x-Then-find-df-dg-

Question Number 211502 by MathematicalUser2357 last updated on 11/Sep/24

If {\begin{cases} f (x) = x^{2} \\ g (x) = \sin x \end{cases},

Then find \frac{d f}{d g} .

Answered by a.lgnaoui last updated on 11/Sep/24

\frac{df}{dg} = \frac{df}{dx} \times \frac{dx}{dg} . = \frac{f^{^{'}}}{g^{'}}

f^{'} = 2 x g^{'} = \cos x

\Rightarrow \frac{df}{dg} = \frac{2 x}{\cos x}

Commented by MathematicalUser2357 last updated on 12/Sep/24

Can be solved in other way .

\frac{d f}{d g} = \frac{2 x d x}{\cos x d x} = \frac{2 x}{\cos x} = 2 x \sec x (No need to transform \frac{1}{\cos x} to \sec x)

Answered by MATHEMATICSAM last updated on 11/Sep/24

f (x) = x^{2} \Rightarrow \frac{d f}{d x} = 2 x

g (x) = \sin x \Rightarrow \frac{d g}{d x} = \cos x

\frac{d f}{d g} = \frac{\frac{d f}{d x}}{\frac{d g}{d x}} = \frac{2 x}{\cos x} = 2 x \sec x

Commented by MathematicalUser2357 last updated on 12/Sep/24