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, x2y+4x3y=1 defines y implicitly. Applying the chain rule to a function of y ddx(4y3) -> ddy(4y3) dydx -> 12y2 dydx -> Applying Chain Rule -> Chain rule More than one dependent variable If y and z both depend on x, then (d)/(dx)(2y3-7ln z+3x2)=6y2(dy)/(dx) -(7)/(z)(dz)/(dx)+6x - Differentiate both sides with respect to x. - Attach dydx or dzdx when the chain rule is used. - Collect the required derivative terms before solving for the derivative.

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 \(y\)

\(\dfrac{\mathrm{d}}{\mathrm{d}x}(4y^3)\)

\(\dfrac{\mathrm{d}}{\mathrm{d}y}(4y^3)\dfrac{\mathrm{d}y}{\mathrm{d}x}\)

\(12y^2\dfrac{\mathrm{d}y}{\mathrm{d}x}\)

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.

Implicit Differentiation

Implicit Differentiation

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