The derivative of a function has many applications to problems in calculus. It is useful to solve problems on (a) rate of change (b) slope/equation of tangent and normal lines (c) increasing and decreasing points/intervals (d) natute of points (e) shape of graph (f) curve sketching (g) approximation of function (h) maximum and minimum problems (i) distance; velocity, and acceleration problems

In this section, we use derivative to solve problems on (b) slope/equation of tangent and normal lines. The derivative of a function at a point is the slope of the tangent line at this point. The normal line is defined as the line that is perpendicular to the tangent line at the point of tangency.

A tangent is a straight line that touches a curve at one point. The idea is that the tangent and the curve are both going in exact same direction at point of contact. In a space curve, number of lines may pass touching one point, therefore, a precise definition of tangent is given as below.

Let C be a curve.

Also let A and B are two points on C

when, point B is moved toward A then limiting form of secant AB, is called tangent to the curve C at A.

Drag point A | Drag point B |

#### Equation of tangent line

Let y=f(x) be a curve C

Also let \(P(x_1,y_1)\) be a given point on C.

Then

Equation of straight line passing through P is

\( y-y_1=m(x-x_1)\)

Since, the tangent line has slope \(m=\frac{dy}{dx}\), the equation of tangent line at P is

\( y-y_1=\frac{dy}{dx} (x-x_1)\)

or
\( y=y_1+\frac{dy}{dx} (x-x_1)\)

This completes the proof

#### Examples : Activity 1

- Find the slope of tangent to the curve \(y=x^3-x\) at x=2
- Find the equation of tangent line to
- to the parabola \(y=x^2\) at the point (1,1)
- to the hyperbola \(y=\frac{3}{x}\) at the point (3,1)
- to the function \(y=\sqrt{x}\) at the point (1,1), (4,2) and (9,3)
- to the parabola \(y=x^2+2x\) at (-3,3)
- to the parabola \(y=x^2-8x+9\) at (3,-6)
- to the curve \(y=(x^2-1)(x-2)\) at the points where the curve cuts the x-axis

- Find the equation of the tangent and normal line to the graph of \(f(x)=\sqrt{x^2+3}\) at the point (−1,2).

#### Examples : Activity 2

- Find the tangent to a curve \(y(x-2)(x-3)-x+7=0\) at a point where it meets x-axis

- Find the horizental and vertical tangent to a curve \(x^2+y^2-xy-27=0\)

- Find the the points on a curve \(y=x^2+3x+4\), the tangents at which pass through the origin

- Find the point at which the tangent to the curve \(y=\sqrt{4x-3}-1\) has its slope \(\frac{2}{3}\)
- Find the equation of all lines having slope 2 and being tangent to the curve \(y+\frac{2}{x-3}=0\). Also find the equations of normal at the points of contact.
- Find the points on the curve \(y=x^3-11x+5\) at which equation of tangent is \(y=x-11\)
- Find the equation of tangent and normal to the curves \(x=a \sin^3 \theta\) and \(y=a \cos^3 \theta\) at \(\theta =\frac{\pi}{4}\)
- If a tangent line to \(y=f(x)\) at (4,3) passes through the point (0,2), find f(4) and f'(4).

#### Examples : Activity 3

Find the tangent and normal to a curve- \(x^2+2y^2=9\) at (1,2)
- \(x^2+y^2=2\) at (1,1)
- \(xy+2x-5y=2\) at (3,2)
- \((y-x)^2=2x+4\) at (6,2)
- \(x+\sqrt{xy}=6\) at (4,1)

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