The Quotient Rule and the Other Trig Functions

The product rule explains the shape. Write $q = \frac{f}{g}$, so $f = qg$. Taking for granted that $q$ is differentiable (a limit argument shows it), the product rule gives $f' = q'g + qg'$. Solve for $q'$:

The order in the numerator matters, because of the minus sign. The term with $f'$ comes first. Swapping the two terms flips the sign of the answer, and it's an easy swap to make.

When the numerator is a constant or a polynomial and the denominator is a single power of $x$, rewriting is quicker. For $y = \frac{5}{x^3}$, write $y = 5x^{-3}$, so $\frac{dy}{dx} = -15x^{-4}$. The quotient rule gives the same answer in more steps.

A constant factor can come out first. For $y = \frac{x^2}{3\sin x}$, write $y = \frac{1}{3} \cdot \frac{x^2}{\sin x}$ and take $\frac{1}{3}$ of the quotient rule's answer, $\frac{2x\sin x - x^2\cos x}{\sin^2 x}$. Left inside, $3\sin x$ has derivative $3\cos x$, and the denominator squared is $9\sin^2 x$.

Each comes from writing the function with sine and cosine and using the quotient rule. For the tangent,

For the cosecant, the numerator $1$ has derivative $0$:

The same steps on $\frac{\cos x}{\sin x}$ and $\frac{1}{\cos x}$ give $-\csc^2 x$ and $\sec x \tan x$. The derivatives of the three co-functions, cosine, cotangent, and cosecant, all carry a minus sign.