IIT JEE Main Maths -Unit 7- Tangents, Normals, and Approximation- Study Notes-New Syllabus

Applications of Derivatives — Tangents, Normals, and Approximation

Derivatives are powerful tools to find the slope of curves, the equation of tangents and normals at specific points, and for approximating values of functions near a given point.

If \( y = f(x) \), then the derivative \( \dfrac{dy}{dx} \) at a point \( x = a \) gives the slope of the tangent to the curve at that point.

Tangent to a Curve

The tangent line to a curve at a point \( P(x_1, y_1) \) is the straight line that just touches the curve at that point and has the same slope as the curve there.

Equation of tangent to \( y = f(x) \) at \( x = x_1 \):

$ y – y_1 = f'(x_1)(x – x_1) $

Here, \( f'(x_1) \) = slope of tangent.

 Normal to a Curve

The normal line to a curve at a point is the line perpendicular to the tangent at that point.

Equation of normal:

$ y – y_1 = -\dfrac{1}{f'(x_1)}(x – x_1) $

Here, slope of normal = \( -\dfrac{1}{\text{slope of tangent}} \).

For Implicit Functions

If the curve is given implicitly as \( F(x, y) = 0 \), then:

$ \dfrac{dy}{dx} = -\dfrac{\dfrac{\partial F}{\partial x}}{\dfrac{\partial F}{\partial y}} $

This value can be used in tangent or normal equations at the desired point.

Approximation using Derivatives

If \( y = f(x) \), and \( x \) changes slightly from \( a \) to \( a + \Delta x \), then:

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

This is used for estimating function values when \( \Delta x \) is small — a key application of the derivative in linear approximation.

 Important Special Forms

• Circle \( x^2 + y^2 = a^2 \)
  Tangent at \( (x_1, y_1) \): \( xx_1 + yy_1 = a^2 \)
  Normal: \( x_1x + y_1y = x_1^2 + y_1^2 \)
• Parabola \( y^2 = 4ax \)
  Tangent at \( (x_1, y_1) \): \( yy_1 = 2a(x + x_1) \)
  Normal: \( y = -\dfrac{y_1}{2a}(x – x_1) + y_1 \)

 Linear Approximation Summary