Implicit Differentiation

Implicit differentiation is used when \(x\) and \(y\) are related by an equation that is not written with \(y\) isolated. Treat \(y\) as a function of \(x\) and apply the chain rule whenever a term contains \(y\).

For example, \(x^2y+4x^3y=1\) defines \(y\) implicitly.

Applying the chain rule to a function of \(y\)

More than one dependent variable

If \(y\) and \(z\) both depend on \(x\), then

\[\begin{aligned}\frac{\mathrm{d}}{\mathrm{d}x}\left(2y^3-7\ln z+3x^2\right)&=6y^2\frac{\mathrm{d}y}{\mathrm{d}x}\\&\quad-\frac{7}{z}\frac{\mathrm{d}z}{\mathrm{d}x}+6x\end{aligned}\]

• Differentiate both sides with respect to \(x\).
• Attach \(\dfrac{\mathrm{d}y}{\mathrm{d}x}\) or \(\dfrac{\mathrm{d}z}{\mathrm{d}x}\) when the chain rule is used.
• Collect the required derivative terms before solving for the derivative.

Example

For the circle \(x^2+y^2=25\), differentiate both sides with respect to \(x\): \(2x+2y\dfrac{\mathrm dy}{\mathrm dx}=0\), so \(\dfrac{\mathrm dy}{\mathrm dx}=-\dfrac xy\). At \((3,4)\) the gradient is \(-\dfrac34\).