## Calculus (Part-2)-geometric explanation of diﬀerentiation

Almost all online maths tutors like the following concept very much

**Geometric explanation of diﬀerentiation- **If we find the derivative of function f(x) at x = x_{0 }then it is equal to the slope of the tangent to the graph of given function f(x) at the given point [(x_{0}, f(x_{0}))].

But what is a tangent line?

**►**It is not merely a simple line that joins the graph of given function at one point.

**►**It is actually the limit of the secant lines joining points P = [(x_{0}, f(x_{0})] and Q on the graph of f(x) as Q moves very much close to P.

The tangent line contacts the graph of given function at given point [(x_{0}, f(x_{0})] the slope of the tangent line matches the direction of the graph at that point. The tangent line is the straight line that best approximates the graph at that point.

As we are given the graph of given function, we can draw the tangent to this graph easily. Still, we’ll like to make calculations involving the tangent line and so will require a calculative method to explore the tangent line.

We can easily calculate the equation of the tangent line by using the slope-point form of the line. We slope of a line is m and its passing through a point (x_{0},y_{0}) then its equation will be

** y − y _{0} = m(x − x_{0})**

So now we have the formula for the equation of the tangent line. It’s clear that to get an actual equation for the tangent line, we should know the exact coordinates of point P. If we have the value of x_{0 }with us we calculate y_{0 }as

** y = f(x _{0})**

The second thing we must know is the slope of line

** m = f’(x _{0})**

Which we call the derivative of given function f(x).

**Definition: **

The derivative f’(x_{0}) of given function f at x_{0} is equal to the slope of the tangent line to y = f(x) at the point P = (x_{0}, f(x_{0}).

All online maths tutors suggest solving fair amount of questions based on this concept

** Example**Find the tangent line to the following function at z=3 for the given function

*Solution*

We can find the derivative of the given function using basic differentiation as discussed in the previous post

We are already given that z=3 so

Equation of tangent line is

** y − y _{0} = m(x − x_{0}) ** here y

_{0}=R(3)=√7

Putting these values we get equation of tangent line

In my online maths tutors series, I will continue discussing calculus and its concepts