Consider the function y = \dfrac{3}{x}.
By first rewriting it in negative index form, differentiate y.
By using the substitutions u = 3 and v = x, differentiate y using the quotient rule.
Find the value of x for which the gradient is undefined.
Consider the function y = \dfrac{2 x - 5}{5 x - 2}.
Using the substitution u = 2 x - 5, find u'.
Using the substitution v = 5 x - 2, find v'.
Hence find y'.
Is it possible for the derivative of this function to be zero?
Consider the function y = \dfrac{5 x^{2}}{2 x + 8}.
Using the substitution u = 5 x^{2}, find u'.
Using the substitution v = 2 x + 8, find v'.
Hence find y'.
Differentiate the following functions using the quotient rule:
Consider the function f \left( t \right) = \dfrac{\left( 4 t^{2} + 3\right)^{3}}{\left(5 + 2 t\right)^{5}}. Find f' \left( t \right).
Consider the function y = \dfrac{6}{\sqrt{x}} - 5.
Find the gradient function using the quotient rule.
Find the gradient of the function at x = 25.
Find the gradient of the tangent to the curve y = \dfrac{9 x}{4 x + 1} at the point \left(1, \dfrac{9}{5}\right).
Find the values of x such that the gradient of the tangent to the curve y = \dfrac{6 x - 1}{3 x - 1} is - 3.
Find the equation of the tangent to y = \dfrac{x}{x + 4} at the point \left(8, \dfrac{2}{3}\right).
Find the equation of the tangent to y = \dfrac{x^{2} - 1}{x + 3} at the point where x = 4.
Find the value of f' \left( 0 \right) for f \left( x \right) = \dfrac{x}{\sqrt{16 - x^{2}}}.
Find the value of f' \left( 4 \right) for f \left( x \right) = \dfrac{4 x^{7}}{\left(x + 4\right)^{4}}.
Find f' \left( 1 \right) for f \left( x \right) = \dfrac{6}{3 + 3 x^{2}} using technology or otherwise.
Find the value of f' \left( 3 \right) for f \left( x \right) = \dfrac{5 x}{9 + x^{2}} using technology or otherwise.
Consider the function y = \dfrac{1}{9 + x^{2}}.
Differentiate y using the quotient rule.
For what values of x is the function decreasing?
Consider the function y = \dfrac{3 - 4 x}{3 x - 4}.
Differentiate y.
Is it possible for the derivative to be zero?
Solve for the value of x that will make the derivative undefined.
Consider the function y = \dfrac{x + 6}{x - 6}.
Differentiate y.
Is it possible for the derivative to be zero?
Solve for the value of x that will make the derivative undefined.
Is the function increasing or decreasing near this value of x?
Consider the function y = \dfrac{5 x}{3 x - 4}.
Differentiate y.
Is it possible for the derivative to be zero?
Solve for the value of x that will make the derivative undefined.
Is the function increasing or decreasing near this value of x?
Consider the function y = \dfrac{2 x + 3}{2 x - 3}.
Differentiate y.
Is it possible for the derivative to be zero?
Solve for the value of x that will make the derivative undefined.
Is the function increasing or decreasing near this value of x?
Consider the function f \left( n \right) = \dfrac{1}{n + 3} + \dfrac{1}{n - 3}.
Differentiate f \left( n \right).
Is the function increasing or decreasing in the intervals \left(-\infty, - 3 \right),\left( - 3 , 3\right) and \left(3, \infty\right)?
Find f' \left( 4 \right).
Consider the function f \left( x \right) = \dfrac{1}{4 + \sqrt{x}} + \dfrac{1}{4 - \sqrt{x}}.
Find the gradient function.
Is the function increasing or decreasing over its domain?
Differentiate y = \dfrac{x^{2}}{x + 3} and find the value of a if y' = 0 at x = a.
Differentiate y = \dfrac{x^{2} + k}{x^{2} - k} and find the possible values of k given that y' = 1 at x = - 3.
Consider the function g \left( x \right) defined as g \left( x \right) = \dfrac{f \left( x \right)}{x^{3} + 3}, where f \left( x \right) is a function of x.
Given that f \left( 2 \right) = 2 and f' \left( 2 \right) = 6, determine the value of g' \left( 2 \right).
Consider the identity 1 + x + x^{2} + \text{. . .} + x^{n - 1} = \dfrac{x^{n} - 1}{x - 1}, where x \neq 1 and n is a positive integer.
Form an expression for the sum 1 + 2 x + 3 x^{2} + \text{. . .} + \left(n - 1\right) x^{n - 2}.
Hence find the value of 1 + 2 \times 5 + 3 \times 5^{2} + \text{. . .} + 8 \times 5^{7}.