Finding the Derivatives of Tangent, Cotangent, Secant, and/or Cosecant Functions AB & BC
All four derivatives come from the quotient rule applied to sine and cosine: $\frac{d}{dx}[\tan x] = \sec^2 x$, $\frac{d}{dx}[\cot x] = -\csc^2 x$, $\frac{d}{dx}[\sec x] = \sec x \tan x$, and $\frac{d}{dx}[\csc x] = -\csc x \cot x$. In the tangent's derivation the subtraction meets the negative from $\frac{d}{dx}[\cos x]$ and the two combine into the $\cos^2 x + \sin^2 x$ that the Pythagorean identity collapses to 1.
Two error families. The four results get mismatched, so $\sec^2$ is paired with cotangent, $\sec x \tan x$ is reused for cosecant, or the minus sign that belongs to every co-function goes missing. And the quotient rule that produces them is run with its two numerator terms reversed or with the double negative mishandled, which is precisely what turns $\sec^2 x$ into an expression that will not simplify.
The work
3 ways in · any order
Lesson
Finding the Derivatives of Tangent, Cotangent, Secant, and/or Cosecant Functions
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Derives all four remaining trigonometric derivatives from the quotient rule, fixes the pattern that puts a minus sign on every co-function, keeps the tangent-secant and cotangent-cosecant families apart, and combines them with the product and quotient rules.
Diagnostic
10-item topic check
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Ten items spanning the two failure modes of this topic: mismatching the four reciprocal trigonometric derivatives or dropping the co-function minus sign, and mishandling the order or the double negative in the quotient rule that produces them.
Targeted Practice
Drill a single misconception
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Pick one of the failure modes you missed and drill it on its own. The round is adaptive: two correct in a row clears the misconception and moves you to the next.