Differentiation Rules Lesson | Derivative Rules with Examples

Lesson Overview

• What Are Differentiation Rules?
• Differentiation Rules of Different Functions
• Important Derivative Rules
• Derivative Rules Examples
• Derivative Rules Assessment

What Are Differentiation Rules?

Differentiation rules are formulas used to find the derivative of a function. A derivative shows how a function changes as its variable changes. These rules make solving derivatives easier. 

This lesson covers all essential differentiation rules and examples to help you understand and apply them effectively.

Example: 

Let's find the derivative of f(x) = x^3 using the power rule.

1. Recall the power rule
   If f(x) = x^n, then the derivative is f'(x) = n * x^(n-1).
2. Identify the power of x
   In f(x) = x^3, the power of x is 3.
3. Apply the power rule
   Multiply the power (3) by x and subtract 1 from the power:
   f'(x) = 3 * x^(3-1)
4. Simplify the result
   f'(x) = 3 * x^2

Final Answer: The derivative of f(x) = x^3 is f'(x) = 3 x^2.

Differentiation rules help solve derivatives efficiently.

Take This Quiz :

Advanced Differentiation Assessment Test

Differentiation Rules of Different Functions

Different functions follow different rules when differentiating. 

There are specific rules for power functions, polynomials, trigonometric functions, exponential functions, and logarithmic functions. 

The rules simplify the process of finding derivatives.

Take This Quiz :

differentiation basic rules quiz

Important Derivative Rules

Understanding the derivative laws is crucial for differentiating various functions. All derivative rules simplify the process, helping determine the rate of change based on the function's structure. 

Let's understand each rule stepwise with the  help of examples.  

1. Power Rule
   This rule is used when differentiating a function with a power of x.

• Rule: If f(x) = x^n, then f'(x) = n * x^(n-1).
  
• Example:
  • Start with f(x) = x^3.
  • Apply the rule: Multiply by the exponent (3) and subtract 1 from the exponent.
  • f'(x) = 3 * x^(3-1) = 3x^2.

2. Constant Rule
   The derivative of a constant is always 0.

• Rule: If f(x) = c, then f'(x) = 0.
  
• Example:
  • Start with f(x) = 5.
  • Since it is a constant, f'(x) = 0.

3. Sum and Difference Rule
   When adding or subtracting functions, take the derivative of each term separately.

• Rule: If f(x) = g(x) + h(x), then f'(x) = g'(x) + h'(x).
  
• Example:
  • Start with f(x) = x^2 + 3x.
  • Differentiate each term: Derivative of x^2 is 2x, and derivative of 3x is 3.
  • Combine results: f'(x) = 2x + 3.

4. Constant Multiplier Rule
   If a function is multiplied by a constant, multiply the derivative by the same constant.

• Rule: If f(x) = c * g(x), then f'(x) = c * g'(x).
  
• Example:
  • Start with f(x) = 4x^3.
  • Apply the power rule to x^3: f'(x) = 3x^2.
  • Multiply by the constant 4: f'(x) = 4 * 3x^2 = 12x^2.

5. Product Rule
   Used when differentiating the product of two functions.

• Rule: If f(x) = u(x) * v(x), then f'(x) = u'(x)v(x) + u(x)v'(x).
  
• Example:
  • Start with f(x) = x^2 * sin(x).
  • Let u(x) = x^2 and v(x) = sin(x).
  • Derivative of u(x) is 2x, and derivative of v(x) is cos(x).
  • Apply the rule: f'(x) = (2x) * sin(x) + (x^2) * cos(x).

6. Quotient Rule
   Used when differentiating a division of two functions.

• Rule: If f(x) = u(x) / v(x), then f'(x) = [u'(x)v(x) - u(x)v'(x)] / [v(x)]^2.
  
• Example:
  • Start with f(x) = x / sin(x).
  • Let u(x) = x and v(x) = sin(x).
  • Derivative of u(x) is 1, and derivative of v(x) is cos(x).
  • Apply the rule: f'(x) = [1 * sin(x) - x * cos(x)] / [sin(x)]^2.

7. Chain Rule
   Used when differentiating a composite function.

• Rule: If f(x) = g(h(x)), then f'(x) = g'(h(x)) * h'(x).
  
• Example:
  • Start with f(x) = (2x + 1)^3.
  • Let g(x) = x^3 and h(x) = 2x + 1.
  • Derivative of g(x) is 3x^2, and derivative of h(x) is 2.
  • Apply the rule: f'(x) = 3(2x + 1)^2 * 2 = 6(2x + 1)^2.

Take This Quiz :

Differentiation Basic Rules Quiz

Derivative Rules Examples

Here are solved examples of differentiation using important derivative rules.

1. Using the Power Rule

Find the derivative of f(x) = x^4.

• Step 1: Identify the exponent n = 4.
• Step 2: Apply the power rule f'(x) = n * x^(n-1).
• Step 3: Multiply by the exponent: 4 * x^(4-1).
• Answer: f'(x) = 4x^3.

2. Using the Constant Rule

Find the derivative of f(x) = 7.

• Step 1: Recognize that 7 is a constant.
• Step 2: Apply the constant rule f'(x) = 0.
• Answer: f'(x) = 0.

3. Using the Sum Rule

Find the derivative of f(x) = x^3 + 5x^2 - 2x + 4.

• Step 1: Differentiate each term separately.
  • Derivative of x^3: 3x^2.
  • Derivative of 5x^2: 10x.
  • Derivative of -2x: -2.
  • Derivative of 4: 0.
• Step 2: Combine the results.
• Answer: f'(x) = 3x^2 + 10x - 2.

4. Using the Product Rule

Find the derivative of f(x) = x^2 * e^x.

• Step 1: Identify the functions:
  • u(x) = x^2, v(x) = e^x.
• Step 2: Differentiate u(x) and v(x):
  • u'(x) = 2x, v'(x) = e^x.
• Step 3: Apply the product rule:
  • f'(x) = u'(x)v(x) + u(x)v'(x).
  • f'(x) = (2x)(e^x) + (x^2)(e^x).
• Answer: f'(x) = e^x(2x + x^2).

5. Using the Chain Rule

Find the derivative of f(x) = (3x^2 + 2)^5.

• Step 1: Identify the outer and inner functions:
  • Outer function g(u) = u^5, inner function h(x) = 3x^2 + 2.
• Step 2: Differentiate g(u) and h(x):
  • g'(u) = 5u^4, h'(x) = 6x.
• Step 3: Apply the chain rule:
  • f'(x) = g'(h(x)) * h'(x).
  • f'(x) = 5(3x^2 + 2)^4 * 6x.
• Step 4: Simplify the expression:
  • f'(x) = 30x(3x^2 + 2)^4.
• Answer: f'(x) = 30x(3x^2 + 2)^4.

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Differentiation Practice Questions With Answers

Derivative Rules Assessment

1. Differentiate f(x) = x^3 + 5x using the power rule.
2. Differentiate f(x) = 2x * cos(x) using the product rule.
3. Differentiate f(x) = x^3 / (x + 1) using the quotient rule.
4. Differentiate f(x) = e^(x^2) using the chain rule.
5. Differentiate f(x) = x^4 - 3x^2 using the sum rule.