The quotient structure
Consider
\[h(x)=\frac{e^{2x}}{1-x^2}\]
The last operation on the whole expression is division, so the quotient rule supplies the template. Both the numerator and the denominator are composites, so each one needs its own chain rule block.
Step 1: name the parts
\[f(x)=e^{2x}, \qquad g(x)=1-x^2\]
Step 2: mark the blocks
The quotient rule template is
\[h'(x)=\frac{\big[,\text{block for }f,\big]\cdot g-f\cdot\big[,\text{block for }g,\big]}{\big[g(x)\big]^2}\]
Three things must be kept straight:
- The numerator contains two chain rule blocks, one for \(f\) and one for \(g\).
- The order in the numerator is block-of-numerator times denominator, minus numerator times block-of-denominator. Reversing the order flips the sign of the whole answer.
- The denominator of the result is the original denominator squared as a whole, not the derivative of the denominator.
Step 3: expand each block
For \(f(x)=e^{2x}\): outer function \(e^{(;)}`, inner function \(2x\).
\[f'(x)=e^{2x}\cdot 2\]
For \(g(x)=1-x^2\): the outer function is the whole expression treated as a sum, and the composite part is \(x^2\).
\[g'(x)=-2x\]
What this page establishes
A quotient with composite numerator and denominator is a quotient rule problem containing two chain rule blocks. The blocks are marked before anything is expanded, and the denominator of the result is the squared original denominator, not a derivative.