Simple Techniques to Find Derivatives
Are you looking for simple techniques to find derivatives? Derivative is a basic concept in calculus that deals with rate of change of function. It is a branch of mathematics that studies functions. Derivative calculator is a measure of the change in function with respect to other independent variables. It is simply a rate of change of function value as 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 which is dependent on x and 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. That represents a horizontal tangent. A derivative calculator can determine the derivative of a function in a single click of a button.
Techniques of Solving Derivatives
There are many methods that can be used to calculate the derivative of a function. 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
Power rule is used to find 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 with respect to x will be mx^(m-1) . For instance, if f(x) = x4, then f'(x) = 4×3. You can easily solve derivative using 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 with respect to x and their derivative be 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 here f(x)=4x^2 and g(x)=3x its derivative will be using the power rule f’(x)+g’(x) = 12x+3. Similarly if function is y=7x^3-4x^2 its derivative will be
f’(x)-g’(x)= 21x^2 – 12x
Product Rule
Product rule is applied to two functions which are differentiable 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 product rule differentiation of two functions in product form will produce two terms one will have derivative of the first function while multiplying the second function remains as it is.
Second term will have a derivative of the second function with multiplying the first function as it is.
Take an example of 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 derivative of any question using 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 is applicable 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 rule 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 functions derivatives such as exponential or logarithmic can be found by using online derivative calculator
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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 with regard to the variable of interest.
Consider an example x^2+y^2=1 take derivative of it
d/dx(x^2)+d/dx(y^2)= 0 derivative of constant is zero
First term is easily solved answering 2x but the second one will apply 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
Total derivative will be 2x+ 2ydy/dx=0
Higher Order Derivatives
You can find 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
It 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.
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 help 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.
Summary
In this article, we showed you some simple techniques to find derivatives. We noted that derivatives are a basic concept in calculus that deals with rate of change of function. It is a branch of mathematics that studies functions.
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