Differentiation of trigonometric composites

Questions involving differentiation of composite trigonometric functions like sin(f(x)), tan(g(x)), or sec²(h(x)) using the chain rule.

3 questions · Moderate -0.4

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CAIE P2 2024 November Q3
3 marks Moderate -0.3
The function \(\text{f}\) is defined by \(\text{f}(x) = \tan^2\left(\frac{1}{2}x\right)\) for \(0 \leqslant x < \pi\).
  1. Find the exact value of \(\text{f}'\left(\frac{\pi}{3}\right)\). [3]
Edexcel C3 Q1
4 marks Moderate -0.5
Use the derivatives of \(\sin x\) and \(\cos x\) to prove that the derivative of \(\tan x\) is \(\sec^2 x\). [4]
Edexcel C3 Q3
9 marks Moderate -0.3
Differentiate each of the following with respect to \(x\) and simplify your answers.
  1. \(\cot x^2\) [2]
  2. \(x^2 e^{-x}\) [3]
  3. \(\frac{\sin x}{3 + 2\cos x}\) [4]