Differentiate implicit relations and variable powers

Implicit differentiation follows a local branch

Treat the dependent variable as a function throughout the calculation. Differentiate both sides, collect its derivative terms, and check their coefficient before dividing.

An explicit expression gives y entirely in terms of x, as in $y=2/x+3x^2$ or $y=4x/(x-1)$.

The source contrasts these with $y=2/x+3y^2$, $xy=y^2-10$ and $\sqrt{x+1}=y^x+e^y$.

An implicit relation need not give one global function.

On a differentiable local branch $y=y(x)$, the chain rule is $d[g(y)]/dx=(dg/dy)(dy/dx)$.

Collect all derivative terms before dividing, and check whether their coefficient is zero.

Examples 11 and Self-Practice 4 apply this procedure; the Teacher supplies the missing product and chain steps in Example 11(b).

Logarithms separate variable powers

For $u^v$ with variable base and exponent, or complicated products and quotients, the source procedure is: take natural logarithms on both sides, simplify logarithms where possible, then differentiate implicitly with respect to x.

Real logarithms require positive arguments.

Example 12 keeps its printed condition $x,y>0$, and Self-Practice 5 proves the derivative of $x^x$ on $x>0$.

A quotient obtained by solving for $dy/dx$ applies only where its denominator is nonzero; retain the preceding undivided equation at exceptional points.