Derivative of a Function

Introduction

Derivative of a function is an essential process in Calculus. Calculus exists to help us solve large and seemingly impossible mathematical problems and calculations. In calculus, we will see that the derivative of a function, with its counterpart, Integrals holds an essential place. A derivative is practically used to measure the steepness of a given graph and this allows it to measure the slope as well. 

What is the Derivative of a function? 

A derivative of a function gives us the slope of any point on a graph if we draw a tangent on that point, o.e, the slope of the graph. 

Definition 

Let x be a value and f(x) a function of x. Now, let us denote the derivative of that function of x as f’. Now, f’ is a function whose domain consisted of those values of x that such a limit exists.

f’x= h0f(x+h) – f(x)h 

So that means the function f(x) is differentiable at b if f’(b) exists. 

We can use the above formula to find the derivative of any function, but it can get tedious, and the sum can get larger than we want it to be. 

Example 

Let us find the derivative of the function f(x) using this formula where f(x)= x. 

First we have to substitute f(x+h) with x+h and f(x) with .x 

That way the formula becomes 

f'(x) = h0x+h-fxh

            = h0x+h-fxh x+h + xx+h + x

            = h0hh(x+h + x)

            = h01(x+h + x)

            Evaluating this limit, we get 1/2x.

So the derivative of the function x is 1/2x. 

This was a smaller function, but this calculation takes a lot of time for larger functions. So we usually go for the d/dx method of differentiating, and for that, we need to keep some formulas prepared, including trigonometric equations. 

In this case, the formula looks like f’(x) = ddxf(x)

Properties of Derivative

The properties of a derivative can be looked at from five different perspectives. 

• The sum of derivatives has to be equal to the derivative of their total. Let us consider two functions f(x) and g(x). So, according to this property 

ddxf(x) + g(x)= ddxf(x) + ddxg(x)

If f(x) is u and g(x) is v then (u+v)’ =  u’ + v’

• In a similar way, the difference of the derivatives of two functions is equal to the derivative of the difference between the functions. Using the same symbols as the previous point we get,

u’ – v’ = (u – v)’ 

• The properties of a derivative include the chain rule of differentiation. Again if we take the functions f(x) and g(x), the chain rule formula will be, 

(fg)(x) = f'(g(x)) g'(x).

• The derivative of a product of two functions can be measured or solved by the product rule of differentiation. 

ddxf(x) g(x) = g(x)ddxf(x) +f(x)ddxg(x)

This rule is also called the LEIBNITZ rule in some books. 

• Similarly, the derivative of the quotient of two functions can be measured by the quotient rule of differentiation. For this rule to work the denominator has to be something other than zero.

ddxf(x)/g(x)= (g(x)df(x)dx-f(x).dg(x)dx)/g(x).g(x)

Roughly these are the properties of differentiation. 

Things to Remember

You have to remember some specific things before attempting to find the derivative of a function. 

• Continuity is an essential factor while calculating the derivative of a function. The derivative of a function only exists if it is continuous. You cannot differentiate a constant. For example if f(x) = 5,  f’(x) will be 0.
• A function will not be differentiable if a tangent at that point turns out to be a vertical line. 
• There are multiple complicated ways for which a function may fail to be differentiable.
• In order to be differentiated at any point,  the function at that point has to be smooth. 

Differentiation of a few trigonometric functions

If f(x) = sin x, f'(x) = cos x

If f(x)= cos x, f'(x) = -sin x

If f(x) = tan x, f'(x) = sec2 x

If f(x) = sec x, f'(x)= sec x tan x

If f(x) = cosec x, f'(x)= -cosec x cot x

If f(x) = cot x, f'(x)= -cosec2 x 

Conclusion

This article covers a whole arena of differentiation including the practical importance and the things to remember while attending tricky short questions. It provides the student with a list of trigonometric functions that one should memorise before attempting to find the derivative of more complicated trigonometric functions.