11.18 Differentiating trigonometric functions
Differentiate the following functions:
Differentiate the following functions:
Differentiate the function x y = \sin 5 x.
Consider \tan x = \dfrac{\sin x}{\cos x} and differentiate y = \tan x using the quotient rule.
Explain why the gradient of the function y = \sin ^{2}\left( 5 x\right) + \cos ^{2}\left( 5 x\right) equal to 0 for all x.
Consider the graph of y = - \sin x which is the gradient function of y = \cos x. A number of points have been labelled on the graph.
Name the point on the gradient function that corresponds to the following locations on the graph of y = \cos x :
Where y = \cos x is increasing most rapidly.
Where y = \cos x is decreasing most rapidly.
Where y = \cos x is stationary.
There is an expansion system in mathematics that allows a function to be written in terms of powers of x. The value of \sin x and \cos x, for any value of x, can be given by the expansions below:
\sin x = x - \dfrac{x^{3}}{3!} + \dfrac{x^{5}}{5!} - \dfrac{x^{7}}{7!} + \dfrac{x^{9}}{9!} - \ldots
\cos x = 1 - \dfrac{x^{2}}{2!} + \dfrac{x^{4}}{4!} - \dfrac{x^{6}}{6!} + \dfrac{x^{8}}{8!} - \ldots
Use the expansions to find:
\dfrac{d}{dx} \left(\sin x\right) in terms of x.
\dfrac{d}{dx} \left(\cos x\right) in terms of x.
Hence express the derivatives of \sin x and \cos x in simplest form.