Why Quotients Need Care
A quotient of trigonometric functions can often be written in a shorter form after differentiation. For example, differentiating \(\frac{\sin x}{\cos x}\) produces an expression that simplifies to \(\frac{1}{\cos^2 x}\). But the original quotient is defined only where \(\cos x\ne 0\). That restriction remains important even when the simplified derivative looks familiar.
The method has two parts: first apply the quotient rule without changing the order of its terms, then simplify using identities. Throughout this tutorial, angles are measured in radians, as required for the standard trigonometric derivative formulas.
The Quotient Rule for Trigonometric Functions
If \(u\) and \(v\) are differentiable functions and \(v(x)\ne 0\), then
For a trigonometric quotient, \(u\) and \(v\) may involve sine, cosine, tangent, or other trigonometric functions. Their derivatives are found using the trigonometric derivative rules and, when the inputs are composite, the chain rule. The denominator \(v(x)\) must be nonzero at the point where the quotient and its derivative are considered.
The order in the numerator matters: it is derivative of the numerator times the denominator, minus the numerator times derivative of the denominator. The entire result is divided by the denominator squared.
Worked Example: Sine Divided by Cosine
Worked Example: Differentiate \(\frac{\sin x}{\cos x}\)
Let \(f(x)=\frac{\sin x}{\cos x}\). The denominator is zero when \(\cos x=0\), so the original function is defined only where \(\cos x\ne 0\).
Set \(u(x)=\sin x\) and \(v(x)=\cos x\). Then \(u'(x)=\cos x\) and \(v'(x)=-\sin x\). Apply the quotient rule:
The identity \(\sin^2 x+\cos^2 x=1\) gives the last simplification. The derivative is valid for \(\cos x\ne 0\). As a check, \(\frac{\sin x}{\cos x}=\tan x\) on that domain, and the known derivative of \(\tan x\) is \(\frac{1}{\cos^2 x}\), with the same restriction.
Why Simplification Must Respect the Domain
An algebraic identity can show that two expressions have equal values wherever both expressions are defined. It does not, by itself, change the domain on which the original function was defined. This distinction is especially visible when simplification cancels a factor that was part of the original denominator.
Here is the relevant reasoning. Suppose two functions agree on an open interval around \(x_0\), and both are differentiable there. Their difference is zero throughout that interval. By the definition of the derivative, the difference of their derivatives at \(x_0\) is therefore zero. Thus their derivatives agree at \(x_0\). The interval condition matters: equality at just one input would not establish equality of derivatives.
Find the inputs where it is zero; those inputs are excluded from the original quotient's domain.
Use the original numerator and denominator, preserving the subtraction order.
Use identities to simplify, but state that the derivative applies only on the original domain.
Worked Example: A Composite Quotient
Worked Example: Differentiate \(\frac{\cos(2x)}{\sin(2x)}\)
Let \(g(x)=\frac{\cos(2x)}{\sin(2x)}\). The denominator is nonzero only when \(\sin(2x)\ne 0\). Set \(u(x)=\cos(2x)\) and \(v(x)=\sin(2x)\). By the chain rule,
Now substitute into the quotient rule:
The identity \(\sin^2(2x)+\cos^2(2x)=1\) justifies the last line. The result is valid where \(\sin(2x)\ne 0\). As a check, the quotient is \(\cot(2x)\) on exactly that domain, whose chain-rule derivative is \(-2\csc^2(2x)=-2/\sin^2(2x)\).
Worked Example: Cancellation and an Excluded Input
Worked Example: Differentiate \(\frac{\sin(2x)}{\cos x}\)
Let \(h(x)=\frac{\sin(2x)}{\cos x}\). The original denominator requires \(\cos x\ne 0\). Take \(u(x)=\sin(2x)\) and \(v(x)=\cos x\), so \(u'(x)=2\cos(2x)\) and \(v'(x)=-\sin x\). The quotient rule gives
Use \(\cos(2x)=\cos^2 x-\sin^2 x\) and \(\sin(2x)=2\sin x\cos x\) in the numerator:
Therefore,
To check, the double-angle identity gives \(h(x)=2\sin x\) wherever the original quotient is defined. Differentiating that equal expression gives \(2\cos x\), in agreement. However, the original quotient still excludes \(\cos x=0\). The formula \(2\cos x\) has values at those points, but those values are not derivatives of the original function there because the original function is not defined there.
A Common Sign Error
A frequent mistake is to reverse the two terms in the quotient-rule numerator. For \(q(x)=\frac{\cos x}{\sin x}\), a wrong calculation might start as follows:
This uses the derivative of the denominator in the first product and the derivative of the numerator in the second; it is not the quotient rule. The correct choices are \(u=\cos x\), \(u'=-\sin x\), \(v=\sin x\), and \(v'=\cos x\). Hence
A practical safeguard is to write \(u'v-uv'\) with the labels still visible before substituting any trigonometric expressions. This makes both the order and the signs easier to check.
Choosing and Checking a Simplified Form
The quotient-rule result is often correct before it is simplified. Simplification is useful because identities may reveal cancellation or a familiar trigonometric derivative. Each algebraic step must still respect the denominator restrictions. In particular, never infer that an excluded input becomes allowed merely because a factor cancels or the final expression has a value there.
For a quotient of trigonometric functions, a reliable final check is to ask:
- Did I identify where the original denominator is zero?
- Did I calculate the derivatives of composite inputs using the chain rule?
- Does the quotient-rule numerator have the order \(u'v-uv'\)?
- Does the identity used in simplification hold for every input under consideration?
- Have I retained the original domain restriction in my final statement?
These checks make the result both algebraically correct and precise about where it applies. Quotient-rule expressions can look complicated, but the underlying method is consistent: identify the domain, differentiate, simplify, and check the domain again before stating the answer.
Check Your Understanding
For each question, pay attention both to the derivative calculation and to the original denominator.
- For \(f(x)=\frac{\sin x}{1+\cos x}\), which inputs are excluded from the domain before differentiating?
- State the quotient rule for \(u(x)/v(x)\), including the condition on \(v(x)\).
- Differentiate \(\frac{\sin(3x)}{\cos(3x)}\). What restriction must accompany your result?
- Why does cancelling a factor in a quotient not necessarily make an excluded input part of the original function's domain?
- For \(q(x)=\frac{\cos x}{\sin x}\), identify \(u,v,u'\), and \(v'\) before applying the quotient rule.
