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Differentiation II

The quotient rule

The quotient rule

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Further kinematics

Vector methods with projectiles

Variable acceleration in one dimension

Differentiating vectors

Integrating vectors

Application of forces

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Statics of rigid bodies

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Motion in two dimensions

Connected particles

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Functions of time: Displacement, velocity, acceleration

Using differentiation to find velocity and acceleration

Maxima and minima problems

Using integration to find velocity and displacement

Constant acceleration formulae

Constant acceleration

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Constant acceleration: Deriving formulae

Constant acceleration formulae: SUVAT

Vertical motion under gravity

Modelling in mechanics

Modelling assumptions

Quantities and units in mechanics

Working with vectors

Normal distribution

The normal distribution

The normal distribution: Finding probabilities

The inverse normal distribution function

The standard normal distribution

Finding the mean and standard deviation

Normal approximation to the binomial distribution

Hypothesis testing with the normal distribution

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The binomial distribution

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Vectors II

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Vectors in 3D

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Applications to mechanics: 3D vectors

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Integration II

Integrating standard functions

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Differentiation II

Differentiating sin x and cos x

Differentiating exponentials and logarithmic functions

The chain rule

The product rule

The quotient rule

Differentiating inverse functions

Differentiating trigonometric functions

Parametric differentiation

Implicit differentiation

Second derivatives: Concave and convex functions

Connected rates of change

Trigonometry and modelling

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Solving trigonometric equations: Double angle formula

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Sequences revision

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Binomial expansion II

Binomial expansion: (1+x)^n

Binomial expansion: (a+bx)^n

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Binomial expansion of compound expressions

Parametric equations

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Using trigonometric identities

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Proof II

Proof by contradiction

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Vectors I

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Integration I

Integrating x^n

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Differentiation I

Finding the gradient of a curve

Differentiation from first principles

Differentiating x^n

Differentiating functions with multiple terms

Gradients, tangents and normals

Increasing and decreasing functions

Second order derivatives

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Binomial expansion I

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Direct and inverse proportion

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Proof I

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Proof by deduction

Proof by exhaustion

Explainer Video

Tutor: Toby

Summary

The quotient rule

In a nutshell

A quotient is the result of dividing one quantity by another. The quotient rule is used to differentiate functions that are fractions consisting of one function on the numerator and one function on the denominator. This is called the quotient of two functions.


Example 1

Identify the two functions that make up the following quotient:

f(x)=3x4−2x+1sin⁡(x)f(x)=\dfrac{3x^4-2x+1}{\sin(x)}f(x)=sin(x)3x4−2x+1​​​


Given that this is a quotient of functions, you look for a function u(x)u(x)u(x) in the numerator and a function v(x)v(x)v(x) in the denominator. So:

u(x)=3x4−2x+1‾u(x)=\underline{3x^4-2x+1}u(x)=3x4−2x+1​ and v(x)=sin⁡(x)‾v(x)=\underline{\sin(x)}v(x)=sin(x)​.


Example 2

Identify the two functions that make up the following quotient:

f(x)=x−3ln⁡(x)f(x)=x^{-3}\ln(x)f(x)=x−3ln(x)​​


Since negative powers result in a reciprocal, you have that:

f(x)=ln⁡(x)x3f(x)=\dfrac{\ln(x)}{x^3}f(x)=x3ln(x)​​​


Thus, you have that the numerator function u(x)u(x)u(x) and denominator function v(x)v(x)v(x) are:

u(x)=ln⁡(x)‾u(x)=\underline{\ln(x)}u(x)=ln(x)​ and v(x)=x3‾v(x)=\underline{x^3}v(x)=x3​.


Note: The function in the second example could alternatively be treated as a product of x−3x^{-3}x−3 and ln⁡(x)\ln(x)ln(x). Hence the product rule can be used to differentiate f(x)f(x)f(x).



The quotient rule

Function f(x)f(x)f(x)​​

Derivative f′(x)f'(x)f′(x)​​

​u(x)v(x)\dfrac{u(x)}{v(x)}v(x)u(x)​​​

​u′(x)v(x)−u(x)v′(x)v(x)2\dfrac{u'(x)v(x)-u(x)v'(x)}{v(x)^2}v(x)2u′(x)v(x)−u(x)v′(x)​​​


PROCEDURE

1.

Given a function f(x)f(x)f(x), identify the numerator function u(x)u(x)u(x) and the denominator function v(x)v(x)v(x).

2.

Differentiate these two each with respect to xxx, so you obtain u′(x)u'(x)u′(x) and v′(x)v'(x)v′(x).

3.

Use the formula f′(x)=u′(x)v(x)−u(x)v′(x)v(x)2f'(x)=\dfrac{u'(x)v(x)-u(x)v'(x)}{v(x)^2}f′(x)=v(x)2u′(x)v(x)−u(x)v′(x)​. Note: The denominator of this is the square of the function v(x)v(x)v(x) and not the composition of vvv with vvv.


Example 3

Differentiate the following function with respect to xxx:

f(x)=x3+22x2f\left(x\right)=\frac{x^3+2}{2x^2}f(x)=2x2x3+2​


Identify the numerator function and denominator function:

u(x)=x3+2u(x)=x^3+2u(x)=x3+2 and v(x)=2x2v(x)=2x^2v(x)=2x2.


Differentiate each of these with respect to xxx:

u′(x)=3x2u'(x)=3x^2u′(x)=3x2 and v′(x)=4xv'(x)=4xv′(x)=4x.


Compute the derivative using the quotient rule:

f′(x)=3x2⋅2x2−(x3+2)⋅4x(2x2)2f^\prime\left(x\right)=\frac{3x^2\cdot2x^2-\left(x^3+2\right)\cdot4x}{\left(2x^2\right)^2}f′(x)=(2x2)23x2⋅2x2−(x3+2)⋅4x​

Simplify:

f′(x)= 6x4−4x4−8x4x4= 2x4−8x4x4= x3−42x3‾\begin{aligned}f'(x)=& \space\dfrac{6x^4-4x^4-8x}{4x^4}\\=&\space\dfrac{2x^4-8x}{4x^4}\\ =&\space \underline{\dfrac{x^3-4}{2x^3}}\end{aligned}f′(x)===​ 4x46x4−4x4−8x​ 4x42x4−8x​ 2x3x3−4​​​




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Differentiating x^n

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Differentiating x^n

Differentiating functions with multiple terms

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Differentiating functions with multiple terms

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The quotient rule

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The quotient rule

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FAQs - Frequently Asked Questions

What is the quotient rule?

The quotient rule says that differentiating f(x)=u(x)/v(x) gives f'(x)={u'(x)v(x)-u(x)v'(x)}/{v(x)^2}.

What is a quotient?

A quotient is the result of dividing one quantity by another.

What type of function can the quotient rule be used on?

Functions of the form f(x)=u(x)/v(x) .

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