Introduction

A derivative is a basic concept of calculus that deals with the rate of change of a function. It is a branch of mathematics that studies functions. A derivative calculator is a measure of the change in function concerning other independent variables. It is simply a rate of change of function value as the input variable changes. a derivative calculator determines the derivative of a function in a few seconds. 

Graphical meaning of Derivative

The derivative of a function is denoted by f’(x) or dy/dx which means y is some function that is dependent on x and the function is denoted by f(x). In terms of geometry, derivatives represent the slope of a tangent line to the graph of a function at a specific point. If the derivative is positive at a particular point the graph is said to increase at that point. If the derivative is negative it means the graph is decreasing. If the derivative of a function is zero, the graph will be a horizontal tangent at that point. The tangent at that point will be horizontal. A derivative calculator can determine the derivative of a function with a single click of a button. 

Techniques of Solving Derivatives

The derivative of a function can be calculated using various techniques. Some common rules include power rule, product rule, quotient rule, chain rule, trigonometric rule, and exponential function rules. In calculus several techniques are used to perform differentiation Some are mentioned below:

Power Rule

The power rule is used to find the differentiation of a function if a variable has some power and in general, we can write f(x)= x^m where m is a constant. The derivative concerning x will be mx^(m-1). For instance, if f(x) = x4, then f'(x) = 4x3. You can easily solve derivatives using a derivative calculator.

Sum and Difference Rule

According to this rule if there are two or more derivatives then their sum after the derivative will be the same as solved separately. Any solution of the derivative can be found by the derivative calculator. Consider two functions f(x) and g(x) both are differentiable concerning x and their derivative is f’(x)+g’(x). This rule tells us how to take derivatives of two or more terms if they are in sum or subtraction form. Mathematically we can write:

f(x)+g(x)= f’(x)+g’(x)

f(x)-g(x)=f’(x)-g’(x)

Consider an example 4x^2 +3x as a function where f(x)=4x^2 and g(x)=3x its derivative will be using the power rule f’(x)+g’(x) = 12x+3. Similarly, if the function is y=7x^3-4x^2 its derivative will be 

f’(x)-g’(x)= 21x^2 - 12x

Product Rule

The product rule is applied to two differentiable functions consider two functions f(x) and g(x) then its derivative will be 

f(x)*g(x)= f’(x)*g(x) +f(x)*g’(x)

According to the product rule differentiation of two functions in product form will produce two terms one will have a derivative of the first function while multiplying the second function remains as it is.

The second term will have a derivative of the second function by multiplying the first function as it is.  

Take an example of a function applying product rule:

 fx * gx= x^4 *(x^2+1)

 f’(x)= 4x^3 , g’(x)=2x 

 f(x) = x^4 , g(x)= (x^2+1)

Formula is  f(x)*g(x)= f'(x)*g(x) +f(x)*g'(x)

Answer = 4x^3*(x^2+1) + x^4*(2x)

You can easily find the derivative of any question using the online tool derivative calculator 

Quotient Rule

This rule allows you to differentiate two functions in division form. If f(x) and g(x) are functions and g(x) is not equal 0, then the derivative is given as

fx/gx= (f’x*gx -fx*g’x) / (gx)^2

Consider an example: d/dx[(5x+1)/(3x-4)] its derivative will be 

f(x)=5x+1 and f’(x)=5

g(x)=3x+4 and g’(x)=3

g(x)^2 = (3x+4)^2

According to formula

= 5*(3x+4 ) - 3(5x+1) / (3x+4)^2

= 17 /(3x+4)^2

Instead of solving complex derivatives, you can use a derivative calculator to find online solutions of derivatives. 

Chain Rule

The chain rule applies to composite functions which means a function is inserted into another function. Given the function y=f(g(x)) depends on g(x) function, its derivative is given by

 f’(gx)= f’(gx) * g’(x)

If y=ln(u) its derivative will be 1/u * du/dx where u is x^2+3x

du/dx= 2x+3 its derivative will be

y’= ln(x^2+3x) = 1/x^2+3x *(2x+3) 

Trigonometric Function Rule:

There are some rules fixed for trigonometric functions like sin(x), cos(x), and tan(x). Consider an example of y’=d/dx(sin(x)) its y’=cos(x)

But y’=d/dx(cosx)= -sin(x)

Exponential and Logarithmic Function:

There are different rules for exponential and logarithmic functions.

 For exponential d/dx(e^x)=e^x but if 

y’=e^u * (du/dx) e.g 

y= e^2x its differentiation 

y’= e^2x (d/dx(2x))

y’= 2*e^2x 

For logarithm 

d/dx(a^x)= a^x*lna

d/dx(a^u)= a^u*lna *du/dx where u is a function

Most complex function derivatives such as exponential or logarithmic can be found by using an online derivative calculator  

Implicit Differentiation

When you have an equation that specifies a function implicitly rather than formally, you utilize implicit differentiation. You solve for the derivative by differentiating both sides of the equation about the variable of interest.

Consider an example x^2+y^2=1 take the derivative of it 

d/dx(x^2)+d/dx(y^2)= 0 derivative of constant is zero

The first term is easily solved by answering 2x but the second one will apply the chain rule as there are two variables x and y. Second term derivative will be d/dy(y^2)*dy/dx simplify it 2ydy/dx

The total derivative will be 2x+ 2ydy/dx=0 

Higher Order Derivatives

You can find the differentiation of a function as many times as its degree for higher-order derivatives. Higher-order derivatives can be represented by f^(n)(x) where n is a natural number. 

Finding the third derivative of a function 5x^3-3x^2+10x-5 

Its first derivative will be f’(x)= 15x^2 - 6x +10

Its second derivative will be f’’(x)=30x-6

Its third derivative will be f’’’(x)=30

A high-order derivative can be found by using an online derivative calculator which is an online math tool to solve derivatives. 

Conclusion

These are some common rules used in mathematics for differentiation. There are other advanced techniques for finding derivatives of multiple variables involved. We can master these techniques by practicing more questions of differentiation. Derivation provides a deeper insight on the rate of change of things which helps us in understanding dynamic systems. They are fundamental blocks in calculus and advanced mathematical techniques. A derivative calculator can be used to solve complex or simple functions by derivation technique. A calculator implements these techniques to find derivatives of a function.