Calculate The Derivative of The Following Function Chegg

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

Basic Rules of Differentiation

Differentiation is the process of finding the derivative of a function. The derivative represents the rate at which a function's value changes with respect to changes in its input variable. There are several basic rules for differentiation:

Power Rule
Product Rule
Quotient Rule
Chain Rule

Each of these rules has specific conditions under which they can be applied. Understanding these rules is essential for calculating derivatives of complex functions.

Power Rule

The Power Rule is used to find the derivative of a function of the form \( f(x) = x^n \), where \( n \) is a constant. The rule states:

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

For example, if \( f(x) = x^3 \), then \( f'(x) = 3x^2 \).

Product Rule

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

\( f'(x) = u'(x) \cdot v(x) + u(x) \cdot v'(x) \).

This rule is essential for finding the derivative of functions that are products of other functions.

Quotient Rule

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

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

This rule is particularly useful for functions that are fractions of other functions.

Chain Rule

The Chain Rule is used for composite functions, where one function is nested inside another. If \( f(x) = g(h(x)) \), then:

\( f'(x) = g'(h(x)) \cdot h'(x) \).

This rule allows us to find the derivative of complex functions by breaking them down into simpler components.

Worked Examples

Let's look at some examples of how to apply these differentiation rules.

Example 1: Power Rule

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

Using the Power Rule: \( f'(x) = 4x^3 \).

Example 2: Product Rule

Find the derivative of \( f(x) = x^2 \cdot \sin(x) \).

Let \( u(x) = x^2 \) and \( v(x) = \sin(x) \). Then:

\( u'(x) = 2x \) and \( v'(x) = \cos(x) \).

Using the Product Rule: \( f'(x) = 2x \cdot \sin(x) + x^2 \cdot \cos(x) \).

Frequently Asked Questions

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 find the derivative of a sum of functions?: The derivative of a sum is the sum of the derivatives. If \( f(x) = u(x) + v(x) \), then \( f'(x) = u'(x) + v'(x) \).
What is the derivative of \( e^x \)?: The derivative of \( e^x \) is \( e^x \). This is a special case of the exponential function.