6 Calculate The Following Derivative

Calculating derivatives is a fundamental skill in calculus that helps determine the rate of change of a function. This guide explains the basic rules of differentiation and provides an interactive calculator to compute derivatives of various functions.

Introduction

Derivatives are essential in mathematics, physics, engineering, and economics. They represent the instantaneous rate of change of a function with respect to a variable. Calculating derivatives involves applying specific rules based on the function's form.

This guide covers the fundamental rules of differentiation and provides an interactive calculator to compute derivatives for different types of functions.

Basic Rules of Differentiation

Differentiation follows several fundamental rules that simplify the process of finding derivatives. These rules include:

Power Rule
Product Rule
Quotient Rule
Chain Rule

Each rule applies to specific types of functions and is essential for solving complex differentiation problems.

Power Rule

The Power Rule is used to differentiate functions of the form \( f(x) = x^n \), where \( n \) is a constant.

If \( f(x) = x^n \), then \( f'(x) = n \cdot x^{n-1} \)

For example, the derivative of \( f(x) = x^3 \) is \( f'(x) = 3x^2 \).

Product Rule

The Product Rule is used when differentiating the product of two functions, \( u(x) \cdot v(x) \).

If \( f(x) = u(x) \cdot v(x) \), then \( f'(x) = u'(x) \cdot v(x) + u(x) \cdot v'(x) \)

For example, the derivative of \( f(x) = x^2 \cdot \sin(x) \) is \( f'(x) = 2x \cdot \sin(x) + x^2 \cdot \cos(x) \).

Quotient Rule

The Quotient Rule is used when differentiating the ratio of two functions, \( \frac{u(x)}{v(x)} \).

If \( f(x) = \frac{u(x)}{v(x)} \), then \( f'(x) = \frac{u'(x) \cdot v(x) - u(x) \cdot v'(x)}{[v(x)]^2} \)

For example, the derivative of \( f(x) = \frac{x^2}{x + 1} \) is \( f'(x) = \frac{2x(x + 1) - x^2}{(x + 1)^2} \).

Chain Rule

The Chain Rule is used for composite functions, where one function is nested inside another.

If \( f(x) = g(h(x)) \), then \( f'(x) = g'(h(x)) \cdot h'(x) \)

For example, the derivative of \( f(x) = \sin(x^2) \) is \( f'(x) = 2x \cdot \cos(x^2) \).

Worked Examples

Let's look at a few examples to illustrate how to apply these rules:

Example 1: Power Rule

Find the derivative of \( f(x) = 4x^5 \).

Using the Power Rule:

\( f'(x) = 5 \cdot 4x^{5-1} = 20x^4 \)

Example 2: Product Rule

Find the derivative of \( f(x) = x \cdot e^x \).

Using the Product Rule:

\( f'(x) = 1 \cdot e^x + x \cdot e^x = e^x (1 + x) \)

Example 3: Chain Rule

Find the derivative of \( f(x) = \cos(3x) \).

Using the Chain Rule:

\( f'(x) = -\sin(3x) \cdot 3 = -3\sin(3x) \)

FAQ

What is the derivative of a constant?

The derivative of any constant is zero. For example, if \( f(x) = 5 \), then \( f'(x) = 0 \).

How do I differentiate \( \sin(x) \) and \( \cos(x) \)?

The derivative of \( \sin(x) \) is \( \cos(x) \), and the derivative of \( \cos(x) \) is \( -\sin(x) \).

What is the difference between the Product Rule and the Chain Rule?

The Product Rule is used for multiplying two functions, while the Chain Rule is used for composing functions (one inside another).