A Level Mathematics Syllabus Statement

Syllabus Content

Differentiate using the product rule, the quotient rule and the chain rule, including problems involving connected rates of change and inverse functions

Here are some exam-style questions on this statement:

"Let $f(x)=\frac{2x}{x^2+3}$" ... more

"Very accurate equipment was used to measure the movement of a particle which moved in a straight line for 3 seconds. Its velocity, $v$ ms^-1 , at time $t$ seconds, is given by:" ... more

"Consider the function $f$ defined by $f(x)= \ln{(x^2 - 9)}$ for $x > 3$." ... more

"If $f(x)=x\sin{x}$, for $-3\le x\le3$" ... more

"A particle moves in a straight line such that its velocity, $v$ ms^-1, at time $t$ seconds is given by:" ... more

"Let $f(x)=\frac{g(x)}{h(x)}$, where $g(3)=36$, $h(3)=12$, $g'(3)=10$ and $h'(3)=4$. Find the equation of the normal to the graph of $f$ at $x=3$." ... more

"The displacement, in millimetres, of a particle from an origin, O, at time t seconds, is given by $s(t) = t^3 cos t + 5t sin t$ where $ 0 \le t \le 5 $ ." ... more

"The edge lengths, $x$ cm, of a cube are increasing at a rate of 6 cm s^-1." ... more

"The function $f$ is defined as follows:" ... more

"Make a sketch of a graph showing the velocity (in $ms^{-1}$) against time of a particle travelling for six seconds according to the equation:" ... more

"The function $f$ is such that $f(x) = \frac{\ln2x}{x^3} $ where $x \gt 0$." ... more

"Let $f(x) = \frac{ln3x}{kx} $ where $ x \gt 0$ and $ k \in \mathbf Q^+ $." ... more

See all these questions

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Calculus It is said that the word calculus comes from the Latin word for the small pebble used for counting and calculations. The two major branches, differentiation and integration, are studied by pupils only towards the end of their school days but does then form a major part of their studies. A course in calculus is a prerequisite for other, more advanced courses in mathematical analysis. Isaac Newton is best known for developing calculus. He was born in England in 1642 and is considered one of the most influential scientists and mathematicians in history.

Furthermore

Examples

1. Chain Rule:
The chain rule is used when differentiating composite functions. If we have a function $ y = f(u) $ and $ u = g(x) $, then the derivative of $ y $ with respect to $ x $ is given by:

$$ \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} $$

Example: Differentiate $ y = \sin(3x^2) $ with respect to $ x $.

Let $ u = 3x^2 $. Then, $ \frac{du}{dx} = 6x $ and $ \frac{dy}{du} = \cos(u) $. Using the chain rule:

$$ \frac{dy}{dx} = \cos(3x^2) \cdot 6x = 6x \cos(3x^2) $$

2. Product Rule:
The product rule is used when differentiating the product of two functions. If $ y = u \cdot v $ where both $ u $ and $ v $ are functions of $ x $, then the derivative of $ y $ with respect to $ x $ is:

$$ \frac{dy}{dx} = u' \cdot v + u \cdot v' $$

Example: Differentiate $ y = x^2 \cdot \ln(x) $ with respect to $ x $.

Using the product rule:

$$ \frac{dy}{dx} = 2x \cdot \ln(x) + x^2 \cdot \frac{1}{x} = 2x \ln(x) + x $$

3. Quotient Rule:
The quotient rule is used when differentiating the quotient of two functions. If $ y = \frac{u}{v} $ where both $ u $ and $ v $ are functions of $ x $ and $ v \neq 0 $, then the derivative of $ y $ with respect to $ x $ is:

$$ \frac{dy}{dx} = \frac{u' \cdot v - u \cdot v'}{v^2} $$

Example: Differentiate $ y = \frac{x^2}{\sin(x)} $ with respect to $ x $.

Using the quotient rule:

$$ \frac{dy}{dx} = \frac{2x \cdot \sin(x) - x^2 \cdot \cos(x)}{\sin^2(x)} $$

Here's a 'deep fake' video featuring the images of Arnold Schwarzenegger, Ice Spice and Mr Beast explaining the chain rule for differentiation. While the humans might be fake the maths is real and correct.

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