A derivative is the rate of change of a variable. The slope of the graph determines this. The derivative of a function constitutes the rate of change of one variable with respect to the second variable. An example of the derivative is the rate of change of velocity. Velocity is the derivative of displacement with respect to time. The rate of change of velocity is called acceleration. Hence, acceleration is the derivative of velocity. The rate of change of momentum is the net force. To find the net force on the object, you need to get the derivative of the object’s momentum.  

Derivatives of some functions

  • d/dx (k) = 0, where k is any constant
  • d/dx(x) = 1
  • d/dx(xn) = nxn-1
  • d/dx (kx) = k, where k is any constant
  • d/dx (√x) = ½ (x)2
  • d/dx (1/x) = -1/x2
  • d/dx (log x) = 1/x, x > 0
  • d/dx (ex) = ex
  • d/dx (ax) = ax log a

Basic theorems of derivatives

  1. The derivative of a constant is zero.  If f(x) = k , where k  is a constant.

  2. The derivative of f (x) = xn where n≠ 0 is f ‘(x) = nxn-1 .
  3. The derivative of f(x) = kg(x), where k is a constant, is f ‘ (x)= kg'(x).
  4. The derivative of f(x) = u(x) ± v(x) is f ‘(x) = u'(x)  ± v'(x)

Order in derivatives

Derivatives can be divided into two basic types based on their order. 

  • First-order derivative  

The main role of the first-order derivative is to classify whether the function is increasing or decreasing. It is clarified as an instantaneous rate of change. The prediction of the first-order derivative is made by using the slope of a tangent. 

  • Second-order derivative

This type of derivative is used in determining the shape of the graph for a given function. The given functions are divided in terms of the concavity of the graph. 

Higher-order derivative

The continuous differentiation of a function is called a higher-order derivative of that function. We will learn about higher-order derivatives in the parametric form in a bit. The second derivative of the function is the derivative of the first derivative, and so on. Calculating higher-order (second, third or more high no. of derivatives) is not hard. You have to keep differentiating till you reach your goal. All the methods and operations we use in first-order derivatives are the same in higher-order derivatives. Higher-order derivatives provide the rate of change of the rate of change in a function. Here is an example of a higher-order derivative:

f(x) = y=  x5  + 3×4 – 5×3 – 2×2 – 9x + 15

f’(x) = dy / dx = 5×4 +12×3 – 15×2 – 4x – 9

f’’(x) = d2y / dx2 = 20×3 + 36×2 + 30x – 4   

f’’’(x) =  d3y / dx3 = 60×2 + 72x + 30

And so on.

Parametric equation 

The type of equation that operates an autonomous variable called a parameter (t) is called a parametric equation. Dependent variables are determined as uninterrupted functions of the parameter. They are not dependent on any other variable. For example, y = x3 can be transformed as x = t,  y = t3 for simplification. This gives great productivity while differentiating and integrating curves. Parametric equations are used to describe and represent all types of curves on the plane. They are used when we cannot plot curves on the cartesian plane. It is also used to plot functions like parabola, ellipse and hyperbola. 

Higher-order derivative in parametric form

The formula of higher-order derivative in parametric form is given by – 

dy = dy * dt = y’(t)    

dx    dt    dx    x’(t)

This is the first-order derivative in parametric form. To get the second-order or higher-order derivatives, differentiate the first equation. Higher-order derivative in parametric form means that we can derive many formulas. We can find the equation of a tangent at any given point (t). The transformation of an equation into a parametric form is called parameterization. Here are some examples of higher-order derivatives parametric form – 

  1. The parametric equations of a circle x2 + y2  = a2  are x = a cos θ and y = a sin θ. Therefore, the parametric coordinates of any point lying on the circle x2 + y2  = a2  are (a cos θ, a sin θ).
  2. The parametric equations of a parabola y2 = 4ax are x = at2 and y = 2at where t is the parameter.

Conclusion 

A derivative is the rate of change of a variable. This is determined by the slope of the graph. The continuous differentiation of a function is called a higher-order derivative of that function. Higher-order derivatives provide the rate of change of the rate of change in a function.  Parametric equations are used to describe and represent all types of curves on the plane. It is also used to plot functions like parabola, ellipse and hyperbola. The formula of higher-order derivative in parametric form is: 

dy/dx=dy/dt.dt/dx=y'(t)x'(t)

We can derive many formulas for differentiation by using higher-order derivatives in parametric form.