Differentiate the following functions:
Differentiate the following 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.
Differentiate y = \tan(5x\degree) where x is an angle in degrees.
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.
If y = 4 \cos 3 x, prove that y'' + 9 y = 0.
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.
Consider the function y = \text{cosec } x.
Find the derivative of this function in terms of \sin x and \cos x.
Show that the derivative is equal to - \cot {x } \times \text{cosec }x.
State the values of x for which \dfrac{d y}{d x} is not defined, on the interval 0 \leq x \leq 2 \pi.
Find the derivative of y = \tan \left(\dfrac{x + 2}{x - 2}\right) using the substitution u = \dfrac{x + 2}{x - 2}.
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.