Derivatives

What is Derivative?

This varying rate of change of a function with respect to an independent variable is termed as a Derivative.

The derivative is primarily used when there is some varying quantity, and the rate of change is not constant. The derivative is used to measure the sensitivity of one variable (dependent variable) with respect to another variable (independent variable).

Meaning of Derivative

It refers to the instantaneous rate of change of a quantity with respect to the other. It helps to investigate the moment by moment nature of an amount.

Derivative Example:

Let a car takes ‘t’  seconds to move from a point ‘a’ to ’b’.

Derivatives

But how long will it take to move from point ‘a’ to ‘c’?

Or

How much distance will it cover in ‘t-1’ seconds?

This can be known from the velocity that is as follows:

Velocity (v) = d(x)/d(t)

Where ‘x’ is the distance travelled and ‘t’ is the time taken to cover that distance.

This will give you the distance covered per unit time so that we can analyze any distance covered in any interval of time.

DERIVATIVE IN CALCULUS

Let ‘y’ be a dependent variable and ‘x’ be an independent variable.

Let there be a change in the value of x, that is dx.

This change in x will bring a change in y, let that be dy.

Now to find out the change in y with a unit change in x can be found as follows:

It is called the derivative.

Derivatives in Calculus Examples

Let f(x) be a function whose value varies as the value of x varies

Steps to find the Derivative:

  1. Change x by the smallest possible value and let that be ‘h’ and so the function becomes f(x+h).
  2. Get the change in value of function that is : f(x + h) – f(x)
  3. The rate of change in function f(x) on changing from ‘x’ to ‘x+h’ will be

 

\(\frac{dy}{dx} = \frac{f(x+h) – f(x)}{h}\)

Now d(x) is ignorable because it is considered to be too small.

Calculus-Derivative Example

Let f(x) be a function where \(f(x) = x^{2}\)

The derivative of x2 is 2x means that with every unit change in x, the value of the function becomes twice (2x).

Limits and Derivatives

When dx is made so small that is becoming almost nothing. With Limits, we mean to say that X approaches zero but does not become zero.

Mathematically: means for all real ε > 0 there exists a real δ > 0 such that for all x with 0 < |x − c| < δ, we have |f(x) − L| < ε

Key Concepts

  • To differentiate a power of x that is in the denominator, first express it as a power with a negative exponent. Eg. \(\frac{1}{^{x2}}= x^{-2}\)
  • Derivative rules simplify the process of differentiating polynomial functions.
  • To differentiate a radical, first, express it as a power with a rational exponent

Apply Derivative Rules to Solve an Instantaneous Rate of Change Problem

A skydiver jumps out of a plane from a height of 2200 m. The skydiver’s height above the ground, in meters, after t seconds is represented by the function \(h\left ( t \right )=2200-4.9t^{2}\)  (assuming air resistance is not a factor). How fast is the skydiver falling after 4 s?

Solution

The instantaneous rate of change of the height of the skydiver at any point in time is represented by the derivative of the height function.

\(h\left ( t \right )=2200-4.9t^{2}\)

\(h’\left ( t \right )=0-4.9 \left ( 2t \right )= -9.8t\)

Substitute \(t=4\)  into the derivative function to find the instantaneous rate of change at 4 s.

\(h’\left ( 4 \right )=9.8\left ( 4 \right )=-39.2\)

After 4 s, the skydiver is falling at a rate of 39.2 m/s.


Practise This Question

The left-hand derivative of f(x)=[x]sin(πx) at x= k, k is an integer and [x] = greatest integer, is