\( \DeclareMathOperator{cosec}{cosec} \)

Sign In | Starter Of The Day | Tablesmaster | Fun Maths | Maths Map | Topics | More

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*" ... more^{-1}, at time \(t\) seconds, is given by: - "
*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*" ... more^{-1}, at time \(t\) seconds is given by: - "
*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 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

Click on a topic below for suggested lesson Starters, resources and activities from Transum.

**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:

**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:

**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:

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

Using the quotient rule:

$$ \frac{dy}{dx} = \frac{2x \cdot \sin(x) - x^2 \cdot \cos(x)}{\sin^2(x)} $$How do you teach this topic? Do you have any tips or suggestions for other teachers? It is always useful to receive feedback and helps make these free resources even more useful for Maths teachers anywhere in the world. Click here to enter your comments.