Question

Use implicit differentiation to find the specified derivative.

y = sin x, dx/dy

Answer 1

There are two ways to do this.
First approach:
You can take the implicit derivative with respect to x, then flip it to dx/dy.

$$
\Large{
\frac{d}{dx} (y) = \frac{d}{dx} (\sin x )
\\ \iff \\
\frac{dy}{dx} = \cos x
\\ \iff \\
\frac{dx}{dy} = \frac{1}{\cos x }
\\ \therefore \\
\frac{dx}{dy} = \sec x


}
$$

Second approach
Take the implicit derivative with respect to y


$$
\Large{
\frac{d}{dy} (y) = \frac{d}{dy} (\sin x )
\\ \iff \\
1 = \cos x \cdot \frac{dx}{dy}
\\ \iff \\
\frac{dx}{dy} = \frac{1}{\cos x }
\\ \therefore \\
\frac{dx}{dy} = \sec x


}
$$