Applications of Derivatives Class 12 Notes | Exercise – 13.3

Applications of Derivatives Class 12 Notes has been updated according to the latest syllabus of 2080. It means the notes of applications of derivatives chapter provided in this article contains all the new exercise that has recently been updated. Now you don’t need to go anywhere searching for the notes of this chapter because we are here to serve you. 

  Chapter – 13  
Applications of Derivatives

In the “Applications of Derivatives” chapter of Class 12 Mathematics, students explore various practical uses of derivatives. The chapter covers the rate of change of quantities, where derivatives help measure how one variable changes in relation to another.⁷

Additionally, students learn to identify increasing and decreasing functions, determine maxima and minima of functions, and apply Rolle’s Theorem and the Mean Value Theorem to understand the behavior of functions over intervals.

Applications of Derivatives Class 12 Notes PDF

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Tangents

A tangent to a curve at a given point is a straight line that just touches the curve at that point without crossing it. The slope of the tangent line at a point on the curve gives the instantaneous rate of change of the function at that point, which is the derivative of the function.

If the equation of the curve is $latex y = f(x)$ and $latex (x_1, y_1)$ is the point of tangency on the curve, then the slope of the tangent at that point is given by the derivative of the function at $latex x = x_1$, i.e., $latex f'(x_1)$.

The equation of the tangent line at the point $latex (x_1, y_1)$ can be written as: $latex y – y_1 = f'(x_1)(x – x_1)$

This equation represents the straight line that touches the curve at $latex (x_1, y_1)$ with the slope $latex f'(x_1)$.

Normals

A normal to a curve at a given point is a straight line that is perpendicular to the tangent at that point. The normal represents the direction of zero change in the function at the point of tangency.

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If the slope of the tangent at $latex (x_1, y_1)$ is $latex f'(x_1)$, then the slope of the normal line is the negative reciprocal of the slope of the tangent, i.e., $latex -\frac{1}{f'(x_1)}$.

The equation of the normal line at the point $latex (x_1, y_1)$ can be written as: $latex y – y_1 = -\frac{1}{f'(x_1)}(x – x_1)$.

This equation represents the straight line that is perpendicular to the tangent and passes through the point $latex (x_1, y_1)$.

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