The term derivative is defined as ‘the rate of change of a quantity with respect to a variable’. To solve problems in calculus and differential equations, derivatives are used.

Derivative of a function f(x) with respect to variable x is given as df(x)dx.

What is a Derivative?

A derivative is defined as the ratio of change of a function f(x) with respect to a variable x i.e. ∆f(x)∆x.

Derivatives are used to calculate following:

  • Slope of a line curve or any polynomial
  • Tangent of a curve
  • Normal to a curve
  • Rate of change of quantities
  • Maxima and minima

Rate of Change of Quantities

The value of a quantity varies with change in the variable. For example, velocity is defined as the rate of change of distance with respect to time i.e. velocity=Δdistancetime

Here, the quantity is distance and variable is time. Velocity is the ratio of change in distance with respect to time.

In the form of derivatives, the rate of change of distance s with respect to time t is defined as dsdt. It is true only if distance s is a function of time t i.e. s=f(t).

If two variables are functions of the third variable, then the derivative is defined in another way. We use the Chain Rule to find the derivative on the rate of change of quantity.

Let us assume that two variables x nd y are changing with respect to third variable u. Let x=f(u) and y=f(u), 

First, we find derivatives dxdu and dydu

Using the chain rule, rate of change dydx is given by

dydx=dydudxdu. In this case, dxdu≠0.

Increasing and Decreasing Functions

A quantity changing with a variable is known as the rate of changing. The value of the quantity increases or decreases with respect to the variable.

For example, the velocity of an object increases or decreases with respect to time.

The graph of a function may increase, decrease, or be constant. Let a function f(x) change from points x1 and x2 over interval I, then the graph f(x):

  1. Increases if x2>x1 in f(x2)>f(x1) for all x1,x2∈I
  2. Decreases if x1>x2 in f(x2)<f(x1) for all x1,x2∈I
  3. Constant if fx=c for all x∈I

Tangents and Normals

The tangent of a curve is defined as ‘a straight line that touches that curve at only one point but never intersects it’.

Let us assume that a line is passing through point x0,y0 and having slope m, then the equation of line is 

y-y0=mx-x0

Where slope m=dydx|x0,y0

Or m=f’x0=tan

The normal line is a line that is perpendicular to the tangent line. The relation between the slope of the tangent line and normal line is m1m2=-1, where m1= slope of tangent line and m2= slope of normal.

 So, slope of normal m2=-1m1=-1f’x0

So, an equation of normal line, y-y0=-1m1x-x0

Or, y-y0=-1f’x0x-x0

Approximation

If the value of a function can’t be calculated easily, then we use the approximation method. Let a function y=f(x). A small increment in x is ∆x. So the increment in y is denoted by ∆y. The increment in y is given as

∆y=fx+∆x-f(x) 

  • dx is the differentiation of x. dx is defined as dx=∆x.
  • Similarly, dy is the differentiation of y. It is defined as dy=f’xdx or dy=dydx∆x.

Maxima and Minima

The derivation is used to find the maximum and minimum value of a function. 

  • Let a function be f(x). If at x=c. fc>f(x), then f(c)= local maximum value of function.
  • Again if fc<f(x), then fc= local minimum value of function.

To find the maximum and minimum value of function f(x), we take the first and second derivatives of f(x).

  • If the second derivatives at point c, f”c<0, then point c is called local maxima and fc= local maximum value.
  • If the second derivatives at point c, f”c>0, then point c is called local minima and fc= local minimum value.

Conclusion

Derivatives, used to solve problems in calculus and differential equations, is ‘the rate of change of a quantity with respect to a variable’. They are used to calculate the slope of a line curve or any polynomial, tangent of a curve, normal to a curve, rate of change of quantities, and maxima and minima. The graphs and equations for each case were outlined and explained.