Use differentiation to find gradients, tangents and normals to curves

Calculus: Gradients, Tangents and Normals

1. Differentiation Recap

Remember that the derivative of a function gives the instantaneous rate of change – the slope of the tangent at any point. The most common rules are:

2. Finding the Gradient of a Curve

Given a curve $y = f(x)$, the gradient at a point $x = a$ is simply the derivative evaluated at that point:

$m_{\text{tangent}} = f'(a)$

Think of it like a roller‑coaster track: the derivative tells you how steep the track is at any spot.

3. Tangent Lines

Once you have the slope $m_{\text{tangent}}$, the equation of the tangent line at $(a, f(a))$ is:

$y - f(a) = m_{\text{tangent}}\,(x - a)$

Example: For $y = x^3 - 3x$, find the tangent at $x = 2$.

1. Compute $y' = 3x^2 - 3$.
2. Evaluate at $x = 2$: $y'(2) = 3(4) - 3 = 9$.
3. Point on curve: $f(2) = 8 - 6 = 2$.
4. Equation: $y - 2 = 9\,(x - 2) \;\Rightarrow\; y = 9x - 16$.

4. Normal Lines

The normal line is perpendicular to the tangent. Its slope is the negative reciprocal of the tangent slope:

$m_{\text{normal}} = -\dfrac{1}{m_{\text{tangent}}}$

Using the same example, the normal at $x = 2$ has slope $-1/9$ and equation:

$y - 2 = -\dfrac{1}{9}\,(x - 2) \;\Rightarrow\; y = -\dfrac{1}{9}x + \dfrac{20}{9}$.

5. Example Problems

1. Problem 1: Find the gradient, tangent and normal at $x = 1$ for $y = \ln x$.
2. Problem 2: For $y = \sin x$, write the equation of the tangent at $x = \dfrac{\pi}{4}$.
3. Problem 3: A curve is given by $y = 3x^2 - 2x + 1$. Determine the point where the tangent is horizontal.

Exam Tips

Happy calculating! 🚀