Topic 2.10 Finding the Derivatives of Tangent, Cotangent, Secant, and/or Cosecant Functions: derive and apply the derivatives of the remaining trigonometric functions.

What this topic is asking

The College Board (Topic 2.10) completes the trigonometric derivatives: $\tan x$, $\cot x$, $\sec x$, and $\csc x$. Each one is built from $\sin x$ and $\cos x$ using the quotient rule, so this topic ties together the trig derivatives of Topic 2.7 with the quotient rule of Topic 2.9.

The derivative table

A pattern helps memory: the two "co" functions (cotangent and cosecant) have negative derivatives, mirroring how $\cos x$ has the negative derivative among $\sin x$ and $\cos x$.

Where they come from

Deriving one to trust the rest

The exam sometimes asks you to derive one of these, which proves you understand where the table comes from. The cleanest is $\tan x$, shown in the worked example below; the others follow the same pattern and use the Pythagorean identity $\sin^2 x + \cos^2 x = 1$.

The memory pattern in full

There is a tidy symmetry that makes all six trig derivatives easier to hold in memory. The three "co" functions - cosine, cotangent, and cosecant - all have a negative sign in their derivatives, while sine, tangent, and secant are positive. Moreover, the derivatives pair up: $\frac{d}{dx}[\tan x] = \sec^2 x$ mirrors $\frac{d}{dx}[\cot x] = -\csc^2 x$, and $\frac{d}{dx}[\sec x] = \sec x \tan x$ mirrors $\frac{d}{dx}[\csc x] = -\csc x \cot x$. Notice that each "co" derivative is the same as its partner's but with every function replaced by its co-function and a minus sign attached. If you firmly know the tangent and secant derivatives, you can write the cotangent and cosecant derivatives by this co-function mirror, which halves what you have to memorize and gives you a self-check.

Putting it all together

With Topics 2.5 through 2.10, you can now differentiate any combination of powers, the six trig functions, exponentials and logarithms, using the power, constant-multiple, sum, product and quotient rules. This completes the fundamental differentiation toolkit of Unit 2; the chain rule for compositions comes in Unit 3. On the exam these derivatives rarely appear in isolation - they show up as one factor inside a product or quotient, or as a term in a sum - so the skill being tested is selecting the correct trig derivative quickly and combining it with the right structural rule. A frequent application is finding the slope of a tangent line to a curve like $y = \sec x$ or $y = \tan x$ at a given point, which simply means evaluating the corresponding derivative there and, if asked, writing the tangent-line equation in point-slope form.