Applications of Derivatives: Tangents and Normals | NIMCET

Quick answer: The derivative $\frac{dy}{dx}$ is the rate of change of y with respect to x. At a point $(x_1, y_1)$ on a curve with slope $m = f'(x_1)$, the tangent is $y - y_1 = m(x - x_1)$ and the normal is $y - y_1 = -\frac{1}{m}(x - x_1)$. For small changes, $\Delta y \approx f'(x)\,\Delta x$.

This applies the differentiation rules.

Rate of change

Tangent and normal

Angle between two curves

The angle between curves at an intersection point is the angle between their tangents:

$\tan\theta = \left\lvert\frac{m_1 - m_2}{1 + m_1m_2}\right\rvert$

The curves cut orthogonally if $m_1m_2 = -1$. This is the same formula as the angle between two lines.

Lengths related to tangent and normal

For a point P on y = f(x) with slope m:

Approximations

$f(x + \Delta x) \approx f(x) + f'(x)\,\Delta x$

Worked examples

Example 1: related rates

The radius of a circle grows at 2 cm/s. How fast is the area growing when r = 5 cm?

$\frac{dA}{dt} = 2\pi r\frac{dr}{dt} = 2\pi \cdot 5 \cdot 2 = 20\pi$ cm²/s.

Example 2: tangent and normal

Find the tangent and normal to $y = x^2 + 1$ at $(1, 2)$.

Step 1: $m = 2x = 2$.

Step 2: Tangent: $y - 2 = 2(x - 1)$, so $y = 2x$.

Step 3: Normal: $y - 2 = -\frac{1}{2}(x - 1)$, so $x + 2y = 5$.

Example 3: point with a given slope

Find the point on $y = x^3 - 3x$ where the tangent is parallel to the x-axis.

$3x^2 - 3 = 0$, so $x = \pm 1$: the points are $(1, -2)$ and $(-1, 2)$.

Example 4: angle between curves

Find the angle between $y = x^2$ and $x = y^2$ at $(1, 1)$.

Step 1: For $y = x^2$, $m_1 = 2$. For $x = y^2$, $1 = 2yy'$, so $m_2 = \frac{1}{2}$.

Step 2:

$\tan\theta = \frac{2 - \frac{1}{2}}{1 + 1} = \frac{3}{4}$

So $\theta = \tan^{-1}\frac{3}{4}$. (At the origin the tangents are the two axes, so there the curves cut orthogonally.)

Example 5: approximation

Approximate $\sqrt{25.2}$.

$f(x) = \sqrt{x}$, $f'(25) = \frac{1}{10}$, so $\sqrt{25.2} \approx 5 + 0.1 \times 0.2 = 5.02$.

Summary table

Exam tips

• Write the equation linking the quantities first, then differentiate with respect to t.
• For the normal, take the negative reciprocal of the tangent slope.
• "Parallel to the x-axis" means $f' = 0$; "parallel to the y-axis" means $\frac{dx}{dy} = 0$.

Next, see monotonicity, maxima and minima, or browse the NIMCET Calculus chapter.

Common mistakes

• Using the slope of the tangent for the normal.
• Substituting values before differentiating in related-rates problems.
• Forgetting units such as cm²/s.
• Confusing relative error with percentage error.

Frequently Asked Questions

What is the equation of the tangent to a curve?

At (x₁, y₁), y − y₁ = f′(x₁)(x − x₁).

What is the equation of the normal?

y − y₁ = −(1/f′(x₁))(x − x₁).

When do two curves cut orthogonally?

When the product of their tangent slopes at the intersection point is −1.

How are derivatives used for approximation?

f(x + Δx) ≈ f(x) + f′(x)Δx for small Δx.

What is a related rates problem?

A problem where two quantities change with time and the rate of one is found from the rate of the other using the chain rule.