Derivative Practice Worksheet with Examples and Solutions

Derivative Practice Worksheet with Examples and Solutions

Derivatives are a fundamental concept in calculus, and practicing with examples is essential to master this topic. In this worksheet, we will go through various examples and solutions to help you understand how to find derivatives.

What is a Derivative?

Before we dive into examples, let’s quickly review what a derivative is. A derivative measures the rate of change of a function with respect to its input. It represents the slope of the tangent line to a curve at a given point.

Basic Rules of Differentiation

There are several basic rules of differentiation that you should be familiar with:

• Power Rule: If f(x) = x^n, then f’(x) = nx^(n-1)
• Product Rule: If f(x) = u(x)v(x), then f’(x) = u’(x)v(x) + u(x)v’(x)
• Quotient Rule: If f(x) = u(x)/v(x), then f’(x) = (u’(x)v(x) - u(x)v’(x)) / v(x)^2
• Chain Rule: If f(x) = g(h(x)), then f’(x) = g’(h(x)) * h’(x)

Examples and Solutions

Now, let’s go through some examples and solutions to help you practice finding derivatives.

Example 1: Find the derivative of f(x) = x^2

💡 Note: Use the power rule to find the derivative.

f(x) = x^2 f’(x) = 2x

Example 2: Find the derivative of f(x) = 3x^2 + 2x - 5

💡 Note: Use the sum rule and power rule to find the derivative.

f(x) = 3x^2 + 2x - 5 f’(x) = 6x + 2

Example 3: Find the derivative of f(x) = (x^2 + 1) / (x + 1)

💡 Note: Use the quotient rule to find the derivative.

f(x) = (x^2 + 1) / (x + 1) f’(x) = ((2x)(x + 1) - (x^2 + 1)) / (x + 1)^2

Example 4: Find the derivative of f(x) = sin(x)

💡 Note: Use the chain rule to find the derivative.

f(x) = sin(x) f’(x) = cos(x)

Example 5: Find the derivative of f(x) = e^(2x)

💡 Note: Use the chain rule to find the derivative.

f(x) = e^(2x) f’(x) = 2e^(2x)

In conclusion, finding derivatives is a crucial skill in calculus, and practicing with examples is essential to master this topic. By following the basic rules of differentiation and practicing with various examples, you can become proficient in finding derivatives.