Derivative Formulas in Calculus

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) = lim_h→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

What are Derivative Formulas?

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.

Examples of Derivative Formula

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) = nx^n-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) = e^x, then:
  f'(x) = e^x

Let's discuss all the Formulas related to derivatives in a structured manner.

Basic Derivative Formulas - Derivative Rules in Calculus

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 = nx^n-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)]

List of Derivative Formulas

The derivative formulas for the different functions are listed below:

Exponential and Logarithmic Derivative Formulas

The derivative formulas for the exponential and logarithmic functions are listed below:

• (d/dx) e^x = e^x
• (d/dx) a^x = a^x ln a
• (d/dx) ln x = (1/x)
• (d/dx) log_ax= (1/x lna)

Read More,

• Logarithms
• Derivative of Exponential Functions

Trigonometric Derivative Formulas

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.

Derivative Formula for Inverse 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

Some Other Derivative Formulas

There are some other functions like implicit functions, parametric functions, and higher order derivatives whose derivative formulas are listed below:

Implicit Derivative Formula

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.

Parametric Derivative Formula

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.

Higher Order Derivative Formula

Finding the derivative of a function for more than one time gives the higher-order derivative of a function.

n^th 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.

Applications of 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

• Calculus Formulas
• Differentiation and Integration Formula
• Logarithmic Differentiation

Solved Question on Derivative Formula

Question 1: Find the derivative of x⁵.
Solution:

Let y = x⁵
⇒ y' = (d/dx) [x⁵]
⇒ y' = 5(x^5-1)
⇒ 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 6^x
Solution:

f(x) = 8 . 6^x
⇒ f'(x) = (d/dx) [8 . 6^x]
⇒ f'(x) = 8 . (d/dx) [6^x]
⇒ 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 + e^x
Solution:

f(x) = 5cos^-1x + e^x
⇒ f'(x) = (d/dx)[5cos^-1x + e^x]
⇒ f'(x) = (d/dx)[5cos^-1x] + (d/dx)[e^x]
⇒ f'(x) = 5(d/dx)[cos^-1x] + (d/dx)[e^x]
⇒ f'(x) = 5[-1/√(1 - x²)] + e^x
⇒ f'(x) = [-5/√(1 - x²)] + e^x

Practice Problems on Derivative Formula

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) = 4^x + 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.