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 = −sinx and d(tanx)/dx = sec²x.
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 = x⁻¹ and v = sinx. Then, du/dx = −1/x² and dv/dx = cosx. So, by the Product rule, dy/dx = −sinx/x² + cosx/x.
Find the derivative of y = 8tan³x.
This can be done by using the Chain rule twice. Let y = 8u³ and u = tanx. Then, dy/du = 24u² and du/dx = sec²x. So, dy/dx = 24u²sec²x = 24tan²xsec²x.
Find the derivative of the following function: f(x) = 5x/(1 + x²) − cosx.
This can be done by using the Quotient rule, where u = 5x and v = 1 + x². Then, du/dx = 5 and dv/dx = 2x. So, f'(x) = ((1 + x²)(5) − (5x)(2x))/(1 + x²)² = (5 + 5x² − 10x²)/(1 + x²)² = (5 − 5x²)/(1 + x²)².
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