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Differentiation

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Question 45

Given that x x\,x is measured in radians, prove, from first principles, that the derivative of cos⁡(x)\cos (x)cos(x) is −sin⁡(x)-\sin (x)−sin(x)

a.

You may assume the formula for cos⁡(A±B)\cos (A \pm B)cos(A±B). Show that

cos⁡(x+h)−cos⁡(x)h=cos⁡xcos⁡h−1h−sin⁡xsin⁡hh \frac{\cos(x+h)-\cos(x)}{h}=\cos x\frac{\cos h-1}{h}-\sin x\frac{\sin h}{h} hcos(x+h)−cos(x)​=cosxhcosh−1​−sinxhsinh​
[2]
b.

You may assume that as h→0h \rightarrow 0h→0 sin⁡hh→1\displaystyle \frac{\sin h}{h} \rightarrow 1hsinh​→1 and cos⁡h−1h→0\displaystyle \frac{\cos h - 1}{h} \rightarrow 0hcosh−1​→0. Hence, show that the derivative of cos⁡(x)\cos (x)cos(x) is −sin⁡(x)-\sin (x)−sin(x)

[4]
Markscheme

Differentiation Questions

  1. A Level
  2. /Maths
  3. /Differentiation

717 exam-style questions on Edexcel A Level Maths Differentiation, covering 9.1 Differentiating sin x and cos x, 9.2 Differentiating exponentials and logarithms, 9.3 The Chain Rule, 9.4 The Product Rule, 9.5 The Quotient Rule, 9.6 Differentiating Trigonometric Functions, 9.7 Parametric Differentiation, 9.8 Implicit Differentiation, 9.9 Using Second Derivatives, and 9.10 Rates of Change. Each one has a worked solution and a mark scheme showing where the marks go.

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