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1.7 Differentiation

1.7 Differentiation

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Question 19

By using the identities tan⁡θ=sin⁡θcos⁡θ\displaystyle \tan \theta = \frac{\sin \theta}{\cos \theta}tanθ=cosθsinθ​ and sec⁡θ=1cos⁡θ\displaystyle \sec \theta = \frac{1}{\cos \theta}secθ=cosθ1​, and the standard derivatives of sin⁡θ \sin \theta\,sinθ and cos⁡θ\cos \thetacosθ, prove the following results:

a.
ddθ(tan⁡θ)=sec⁡2θ \frac{d}{d\theta}(\tan \theta) = \sec^2 \theta dθd​(tanθ)=sec2θ
[3]
b.
ddθ(sec⁡θ)=sec⁡θtan⁡θ \frac{d}{d\theta}(\sec \theta) = \sec \theta \tan \theta dθd​(secθ)=secθtanθ
[3]
c.
ddθ(cot⁡θ)=−csc⁡2θ \frac{d}{d\theta}(\cot \theta) = -\csc^2 \theta dθd​(cotθ)=−csc2θ
[3]
d.
ddθ(csc⁡θ)=−csc⁡θcot⁡θ \frac{d}{d\theta}(\csc \theta) = -\csc \theta \cot \theta dθd​(cscθ)=−cscθcotθ
[3]
Markscheme

1.7 Differentiation Questions

  1. A Level
  2. /Maths
  3. /1.7 Differentiation

425 exam-style questions on OCR A Level Maths 1.7 Differentiation, covering 1.7.1 Derivative as gradient of the tangent, 1.7.2 Gradient of the tangent at a point, 1.7.3 Sketching the gradient function, 1.7.4 Second derivatives, 1.7.5 Second derivative as rate of change of gradient, 1.7.6 Convex, concave and points of inflection (A-level only), 1.7.7 Differentiation from first principles for powers of x, 1.7.8 Differentiation from first principles for sin x and cos x (A-level only), 1.7.9 Differentiating x^n, 1.7.10 Differentiating e^(kx) and a^(kx) (A-level only), 1.7.11 Differentiating trigonometric functions (A-level only), 1.7.12 Derivative of ln x (A-level only), 1.7.13 Tangents and normals, 1.7.14 Stationary points, 1.7.15 Increasing and decreasing functions, 1.7.16 Points of inflection (A-level only), 1.7.17 Product and quotient rules (A-level only), 1.7.18 Chain rule (A-level only), 1.7.19 Parametric and implicit differentiation (A-level only), 1.7.20 Constructing differential equations (A-level only), and 1.7 Differentiation. Each one has a worked solution and a mark scheme showing where the marks go.

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