Question

Use implicit differentiation to find dy/dx

cosxy + x^7 = y^7

Answer 1

We have
\[\cos xy+x^7=y^7\]
we know the differentiation of the following
\[\frac{d}{dx} cos x=-sin x\]
and
\[\frac{d}{dx} x^n=nx^{n-1}\]
Let's use these here
\[\frac{d}{dx}(\cos xy +x^7)=\frac{d}{dx} y^7\]
we get
\[-\sin xy \times \frac{d}{dx}(xy)+7x^6=7y^6 \frac{dy}{dx}\]

\[-\sin xy (y+x\frac{dy}{dx})+7x^6=7y^6 \frac{dy}{dx}\]
we get
\[-y \sin xy-x \sin xy \frac{dy}{dx}+7x^6=7y^6 \frac{dy}{dx}\]
or
\[-y \sin xy- 7x^6=7y^6 \frac{dy}{dx}+ \sin xy \frac{dy}{dx}\]
Or we get
\[\frac{dy}{dx}=\frac{-y \sin xy- 7x^6}{\sin xy +7y^6}\]