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Derivative (Differentiation) Calculator
To find the derivative enter function, select variable, write order, and press calculate button using derivative calculator
Table of Contents:
Derivative Calculator
Derivative calculator is used to calculate the derivative of a function. This differentiation calculator differentiates the function step by step.
What is a derivative?
A derivative is used to find the change in a function with respect to the change in a variable. “In mathematics, a derivative is the rate of change of a function with respect to a variable. Derivatives are fundamental to the solution of problems in calculus and differential equations.”
Wikipedia states that,
“The derivative of a function of a real variable measures the sensitivity to change of the output value with respect to a change in its input value.”
Derivative Rules (Techniques)
Below, you will find the basic and advanced derivative rules, which will help you understand the whole process of derivation.
Sum & Difference Rule
According to the sum & difference rule the notation of differentiation will be applied to each term of the function separately.
d/dx [f(x) ± g(x)] = d/dx [f(x)] ± d/dx [g(x)]
Constant rule
The derivative of any constant would be 0 in any case.
d/dx K = 0
Where K is any constant
Product rule
d/dx [f(x) * g(x)] = d/dx [f(x)] * g(x) + f(x) d/dx [g(x)]
If the above equation confuses you, use the product rule calculator above to differentiate a function using the product rule.
Quotient rule
d/dx [f(x) / g(x)] = 1/[g(x)]2 [d/dx [f(x)] * g(x) + f(x) d/dx [g(x)]]
Chain rule
If f(x) = h (g(x))
f'(x) = h' (g(x)).g' (x)
This calculator also acts as a chain rule calculator because it uses the chain rule for derivation whenever it is necessary.
Derivatives cannot be evaluated by using a single static formula. There are specific rules to evaluate each type of function.
| Rule Name | Expression |
| Exponent Rule | ddxex = ex |
| Logarithmic Rules | ddx ax = ax ln(a), a > 0 ddx ln(x) = 1x , x > 0 ddx logx(x) = 1x ln(a) , x , x > 0 |
| Trigonometric Rules | ddx sin(x) = cos(x) ddx cos(x) = -sin(x) ddx tan(x) = sec2(x) = 1cos2(x) = 1 + tan2(x) |
| Inverse Trigonometric Rules | ddx arcsin(x) = 11 - x2 ddx arccos(x) = - 11 - x2 ddx arctan(x) = 11 - x2 |
| Power Rule | ddxxa = ax(a-1) |
Examples of derivatives
Example 1
Find out the derivative of the following function.
f(x) = (x2 + 5)3
Solution:
Step 1: As we can see, the given function can be evaluated by chain rule.
f(x) = (x2 + 5)3
Step 2: Write down the chain rule.
f'(x) = h'(g(x)).g' (x)
Step 3: Let’s apply the chain rule to the given function.
f'(x) = 3(x2 + 5)3-1 f'(x2 + 5)
The left part of the function is evaluated. Now, to solve the right part of the function, we can apply the sum rule because the expression contains the sum operator.
f'(x) = 3(x2 + 5)2 (f'(x2) + f'(5))
f'(x) = 3(x2 + 5)2 ((2x) + (0)) → f'(x) = 0
f'(x) = 6x(x2 + 5)2
Example 2
Solve the derivative of the given function.
f(x) = (x3 - 2) * (x2 + x - 4)
Solution:
Step 1: Here, we will use the product rule to solve the given expression.
f(x) = (x3 - 2) * (x2 + x - 4)
Step 2: Write down the product rule.
d/dx [f(x) * g(x)] = d/dx [f(x)] * g(x) + f(x) d/dx [g(x)]
Step 3: Apply the product rule to solve the expression.
d/dx f(x) = (x2 + x - 4) d/dx [x3 - 2] + (x3 - 2) d/dx [x2 + x - 4]
Differentiate the above expression with the help of sum, difference, constant, and power rules.
d/dx f(x) = (x2 + x - 4) [d/dx(x3) - d/dx(2)] + (x3 - 2) [d/dx(x2) + d/dx(x) - d/dx(4)]
d/dx f(x) = (x2 + x - 4) [3x2 - 0] + (x3 - 2) [2x + 1 - 0]
d/dx f(x) = (x2 + x - 4) [3x2] + (x3 - 2) [2x + 1]
d/dx f(x) = (3x4 + 3x3 - 12x2) + (2x4 + x3 - 4x - 2)
d/dx f(x) = 5x4 + 4x3 - 12x2 - 4x - 2
FAQs
How do you calculate derivatives?
Derivatives can be calculated in several ways according to the function. The derivative of a constant would be zero. There are numerous rules of derivation which we can apply according to the nature of the function, i.e., sum, product, chain rule, etc.
f(x) = x2 + 2x - 3
f'(x) = 2x2-1 + 2(1) - 0
f'(x) = 2x + 2
Why do we calculate derivatives?
We calculate the derivatives to compute the rate of change in one object because of the change in another object. For example, dxdy simply means that we are calculating the total change that occurred in x object due to the change in y object.
What's a derivative in math?
In mathematics, a derivative is the measure of the rate of change with respect to a variable. For example, we can calculate the change in the speed of a car for a specific time period using time as a variable.
References:
- Encyclopædia Britannica, inc. (n.d.). Derivative. Encyclopædia Britannica.
- Calculus I - The Definition of the Derivative by tutorial.math.lamar.edu.