Differentiation: techniques, stationary points, tangents, normals, rates of change

Pure Mathematics 1 – Differentiation 📈

1. Differentiation Techniques 🔍

Differentiation is the art of finding the rate at which a quantity changes. Think of it as the speedometer of a function – it tells you how fast the function is moving at any point.

1. Power Rule: If \(f(x)=x^n\) then \(f'(x)=n\,x^{\,n-1}\). Example: \(f(x)=x^3 \Rightarrow f'(x)=3x^2\).
2. Product Rule: If \(f(x)=u(x)v(x)\) then \(f'(x)=u'(x)v(x)+u(x)v'(x)\). Example: \(f(x)=x^2\sin x \Rightarrow f'(x)=2x\sin x + x^2\cos x\).
3. Quotient Rule: If \(f(x)=\dfrac{u(x)}{v(x)}\) then \(f'(x)=\dfrac{u'(x)v(x)-u(x)v'(x)}{[v(x)]^2}\). Example: \(f(x)=\dfrac{x}{\ln x} \Rightarrow f'(x)=\dfrac{\ln x-1}{(\ln x)^2}\).
4. Chain Rule: If \(f(x)=g(h(x))\) then \(f'(x)=g'(h(x))\,h'(x)\). Example: \(f(x)=\sin(x^2) \Rightarrow f'(x)=\cos(x^2)\cdot 2x\).

2. Stationary Points ⚙️

Stationary points occur where the derivative is zero: \(f'(x)=0\). They can be peaks, valleys, or points of inflection.

Example: Find stationary points of \(f(x)=x^3-3x^2+2\). 1. \(f'(x)=3x^2-6x\). 2. \(3x^2-6x=0 \Rightarrow x=0,\,2\). 3. \(f''(x)=6x-6\). - At \(x=0\): \(f''(0)=-6<0\) → local maximum. - At \(x=2\): \(f''(2)=6>0\) → local minimum.

3. Tangents and Normals ✏️

A tangent line touches a curve at one point and has the same slope as the curve there. The normal is perpendicular to the tangent.

• Tangent line equation: \(y-y_0 = f'(x_0)(x-x_0)\), where \((x_0,y_0)\) is the point of tangency.
• Normal line equation: Slope of normal \(m_n = -\dfrac{1}{f'(x_0)}\). Equation: \(y-y_0 = m_n (x-x_0)\).

Example: Find the tangent and normal at \(x=1\) for \(f(x)=\sqrt{x}\). 1. \(f'(x)=\dfrac{1}{2\sqrt{x}}\). 2. At \(x=1\): \(f'(1)=\dfrac12\). 3. Tangent: \(y-1 = \dfrac12(x-1)\). 4. Normal slope: \(-2\). Normal: \(y-1 = -2(x-1)\).

4. Rates of Change 🚀

Rates of change describe how one quantity changes with respect to another. In physics, this could be speed (change of position over time) or acceleration (change of speed over time).

1. Instantaneous rate of change is the derivative at a point.
2. Average rate of change over an interval \([a,b]\) is \(\dfrac{f(b)-f(a)}{b-a}\).

Example (Speed): Position \(s(t)=t^2+3t\). Speed \(v(t)=s'(t)=2t+3\). At \(t=2\) s, speed is \(v(2)=7\) m/s. Acceleration \(a(t)=v'(t)=2\) m/s² (constant).

🎉 Keep practising! The more you differentiate, the faster you’ll spot patterns and solve problems. Happy maths!