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Differentiation of trigonometric functions for SQA Higher Maths

This page covers the following topics:

1. Differentiating trigonometric functions

Equations involving trigonometric functions can be differentiated using the basic rules of differentiation by using the following results: d(sinx)/dx = cosx, d(cosx)/dx = โˆ’sinx and d(tanx)/dx = secยฒx.

Differentiating trigonometric functions

1

Find a function for the slope for the given graph.

The equation of the graph is given by y = sinx/x. To find a function for the slope, this must be differentiated. This can be done using the Product rule, where u = 1/x = xโปยน and v = sinx. Then, du/dx = โˆ’1/xยฒ and dv/dx = cosx. So, by the Product rule, dy/dx = โˆ’sinx/xยฒ + cosx/x.

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2

Find the derivative of y = 8tanยณx.

This can be done by using the Chain rule twice. Let y = 8uยณ and u = tanx. Then, dy/du = 24uยฒ and du/dx = secยฒx. So, dy/dx = 24uยฒsecยฒx = 24tanยฒxsecยฒx.

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3

Find the derivative of the following function: f(x) = 5x/(1 + xยฒ) โˆ’ cosx.

This can be done by using the Quotient rule, where u = 5x and v = 1 + xยฒ. Then, du/dx = 5 and dv/dx = 2x. So, f'(x) = ((1 + xยฒ)(5) โˆ’ (5x)(2x))/(1 + xยฒ)ยฒ = (5 + 5xยฒ โˆ’ 10xยฒ)/(1 + xยฒ)ยฒ = (5 โˆ’ 5xยฒ)/(1 + xยฒ)ยฒ.

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