Derivative Formulas in Calculus
Last Updated :
22 Apr, 2025
Derivative Formulas in Calculus are one of the important tools of calculus as Derivative formulas are widely used to find derivatives of various functions with ease and also, help us explore various fields of mathematics, engineering, etc.

The derivatives represent the rate of function with respect to any variable. The derivative of a function f(x) is denoted as f'(x) or (d/dx) [f(x)]. The process of finding derivatives is called differentiation.
The most fundamental derivative formula is the definition of a derivative, which is defined as:
f'(x) = limh→0 [(f(x + h) - f(x))/h]
There are various derivative formulas including general derivative formulas, derivative formulas for trigonometric functions, and derivative formulas for inverse trigonometric functions, etc.
Read in Detail: Calculus in Maths
Derivative Formulas are those mathematical expressions which help us calculate the derivative of some specific function with respect to its independent variable. In simple words, the formulas which helps in finding derivatives are called as derivative formulas. There are multiple derivative formulas for different functions.
Some examples of formulas for derivatives are listed as follows:
- Power Rule: If f(x) = xn, where n is a constant, then the derivative is given by:
f'(x) = nxn-1
- Constant Rule: If f(x) = c, where c is a constant, then the derivative is zero:
f'(x) = 0
- Exponential Functions: If f(x) = ex, then:
f'(x) = ex
Let's discuss all the Formulas related to derivatives in a structured manner.
Some of the most basic formulas to find derivatives are:
Let's discuss these rules in detail:
Constant Rule for Derivatives
The constant rule for derivatives is given by:
(d/dx) constant = 0
Power Rule for Derivatives
The Power Rule for derivatives is given by:
(d/dx) xn = nxn-1
Sum Difference Rule for Derivatives
The sum and difference rule for derivatives is given by:
(d/dx) [f(x) ± g(x)] = (d/dx) f(x) ± (d/dx) g(x)
Product Rule for Derivatives
The Product Rule for derivatives is given by:
(d/dx) [f(x). g(x)] = f'(x). g(x) + f(x). g'(x)
Quotient Rule for Derivatives
The Quotient Rule for derivatives is given by:
(d/dx) [f(x)/g(x)] = [g(x).f'(x) - f(x). g'(x)]/[g(x)]2
Chain Rule for Derivatives
The Chain Rule for derivatives is given by:
(d/dx) [f(g(x))] = (d/dx) [f(g(x))] × (d/dx) [g(x)]
The derivative formulas for the different functions are listed below:
The derivative formulas for the exponential and logarithmic functions are listed below:
- (d/dx) ex = ex
- (d/dx) ax = ax ln a
- (d/dx) ln x = (1/x)
- (d/dx) logax= (1/x lna)
Read More,
The derivative formulas for the trigonometric functions are listed below:
- (d/dx) sin x = cos x
- (d/dx) cos x = -sin x
- (d/dx) tan x = sec2 x
- (d/dx) cot x = -cosec2x
- (d/dx) sec x = sec x . tan x
- (d/dx) cosec x = - cosec x . cot x
Learn more about Derivative of Trigonometric Functions.
The derivative formulas for the inverse trigonometric functions are listed below:
- (d/dx) sin-1 x = 1/[√(1 - x2)]
- (d/dx) cos-1 x = -1/[√(1 - x2)]
- (d/dx) tan-1 x = 1/(1 + x2)
- (d/dx) cot-1 x = -1/(1 + x2)
- (d/dx) sec-1 x = 1/[|x|√(x2 - 1)]
- (d/dx) cosec-1 x = -1/[|x|√(x2 - 1)]
Read more, Derivative of Inverse Trig Functions.
Derivative of Hyperbolic Functions
The derivative formulas for the trigonometric functions are listed below:
- (d/dx) sinh x = cosh x
- (d/dx) cosh x = sinh x
- (d/dx) tanh x = sech2 x
- (d/dx) coth x = -cosech2x
- (d/dx) sech x = -sech x . tanh x
- (d/dx) cosech x = -cosech x . coth x
There are some other functions like implicit functions, parametric functions, and higher order derivatives whose derivative formulas are listed below:
The method of finding the derivative of an implicit function is called implicit differentiation. Let's take an example to understand the method of finding derivatives implicitly.
Example: Find the derivative of xy = 2
Solution:
(d/dx) [xy] = (d/dx) 2
⇒ x(dy/dx) + y(dx/dx) = 0
⇒ x(dy/dx) + y(1) = 0
⇒ x(dy/dx) + y = 0
⇒ x(dy/dx) = -y
⇒ (dy/dx) = -y/x
From given equation y = 2/x
(dy/dx) = -(2/x)/x
⇒ (dy/dx) = -(2/x2)
Learn more about Implicit Differentiation.
If the function y(x) is expressed in terms of the third variable t, and x and y can be represented in the form x = f(t) and y = g(t), then this type of function is called a parametric function.
If y is a function of x and x = f(t) and y = g(t) are two differentiable functions of parameter t, then, derivative of a parametric function is given by:
(dy/dx) = (dy/dt)/(dx/dt), such that (dx/dt) ≠ 0
Read more about Parametric Differentiation.
Finding the derivative of a function for more than one time gives the higher-order derivative of a function.
nth Derivative = dny/(dx)n
Read more about Higher Order Derivative.
How to find the Derivatives?
To find the derivatives of a function, we follow the steps below :
- First, check the type of the function, whether it is algebraic, trigonometric, etc.
- After finding the type, apply the corresponding derivative formulas to the function.
- The resultant value gives the derivative of the function using the derivative formula.
There are many applications of the derivative formulas. Some of these applications are listed below:
- Derivatives are used to find the rate of change in any quantity.
- It can be used to find maxima and minima.
- It is used in increasing and decreasing functions.
Related Reads
Question 1: Find the derivative of x5.
Solution:
Let y = x5
⇒ y' = (d/dx) [x5]
⇒ y' = 5(x5-1)
⇒ y' = 5x4
Question 2: Find the derivative of log2x.
Solution:
Let y = log2x
⇒ y' = (d/dx) [log2x]
⇒ y' = 1/ [x ln2]
Question 3: Find the derivative of the function f(x) = 8 6x
Solution:
f(x) = 8 . 6x
⇒ f'(x) = (d/dx) [8 . 6x]
⇒ f'(x) = 8 . (d/dx) [6x]
⇒ f'(x) = 8[6x ln 6]
Question 4: Find the derivative of the function f(x) = 3sinx + 2x
Solution:
f(x) = 3 sinx + 2x
⇒ f'(x) = (d/dx)[3 sinx + 2x]
⇒ f'(x) = (d/dx)[3 sinx] + (d/dx)[2x]
⇒ f'(x) = 3(d/dx)[sinx] + 2(d/dx)(x)
⇒ f'(x) = 3 cosx + 2(1)
⇒ f'(x) = 3 cosx + 2
Question 5: Find the derivative of the function f(x) = 5cos-1x + ex
Solution:
f(x) = 5cos-1x + ex
⇒ f'(x) = (d/dx)[5cos-1x + ex]
⇒ f'(x) = (d/dx)[5cos-1x] + (d/dx)[ex]
⇒ f'(x) = 5(d/dx)[cos-1x] + (d/dx)[ex]
⇒ f'(x) = 5[-1/√(1 - x2)] + ex
⇒ f'(x) = [-5/√(1 - x2)] + ex
Problem 1: Evaluate: (d/dx) [x4].
Problem 2: Find the derivative of y = 5cos x.
Problem 3: Find the derivative of y = cosec x + cot x.
Problem 4: Find the derivative of f(x) = 4x + log3x + tan-1 x.
Problem 5: Evaluate: (d/dx) [40].
Problem 6: Find the derivative of f(x) = x5 + 5x3 + 1.
Summary
Derivative formulas in calculus provide essential tools for finding the rates of change of various functions. These formulas include the power rule, product rule, quotient rule, and chain rule, along with derivatives of common functions like exponential, logarithmic, and trigonometric functions. The power rule states that the derivative of x^n is nx^(n-1). The product rule allows us to differentiate the product of two functions, while the quotient rule helps with ratios of functions. The chain rule is crucial for composite functions. Memorizing and understanding these formulas enables efficient differentiation of complex functions and is fundamental to many applications in mathematics, physics, and engineering.
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