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MathCalculus

Derivative of Trigonometric Functions: Differentiate csc x + x tan x

The sum and product rules give f′(x) = −csc x cot x + tan x + x sec²x on the original function's domain.

Question

Find the derivative of

f(x)=csc⁡x+xtan⁡x.f(x)=\csc x+x\tan x.

Answer

Differentiate the two terms separately:

ddx(csc⁡x)=−csc⁡xcot⁡x.\frac{d}{dx}(\csc x)=-\csc x\cot x.

For xtan⁡xx\tan x, use the product rule:

ddx(xtan⁡x)=(1)(tan⁡x)+x(sec⁡2x)=tan⁡x+xsec⁡2x.\frac{d}{dx}(x\tan x) =(1)(\tan x)+x(\sec^2x) =\tan x+x\sec^2x.

Therefore,

f′(x)=−csc⁡xcot⁡x+tan⁡x+xsec⁡2x.\boxed{f'(x)=-\csc x\cot x+\tan x+x\sec^2x}.

Evidence boundary

The derivative is valid only where the original function is defined, so both sin x and cos x must be nonzero. The result uses radian-based trigonometric derivative formulas.

Sources

These references support the concepts and methods used in the explanation above.