Chegg 6 Calculate The Following Derivative

Calculating derivatives is a fundamental skill in calculus that helps determine rates of change and slopes of curves. This guide explains the Chegg 6 method for calculating derivatives, provides a calculator, and includes examples and FAQs.

Introduction

Derivatives are a core concept in calculus that represent the rate at which a function changes as its input changes. They are essential in physics, engineering, economics, and many other fields. The Chegg 6 method provides a structured approach to calculating derivatives using basic rules and techniques.

This guide will cover the fundamental rules of differentiation, provide a calculator for quick calculations, and include worked examples to help you master this important mathematical concept.

Basic Rules of Differentiation

There are several fundamental rules for differentiating functions:

Power Rule: For a function f(x) = x^n, the derivative is f'(x) = n*x^(n-1).
Product Rule: For two functions u(x) and v(x), the derivative of their product is (u*v)' = u'v + uv'.
Quotient Rule: For two functions u(x) and v(x), the derivative of their quotient is (u/v)' = (u'v - uv')/v².
Chain Rule: For a composite function f(g(x)), the derivative is f'(g(x))*g'(x).

Remember that differentiation is linear, meaning the derivative of a sum is the sum of the derivatives, and the derivative of a constant is zero.

Power Rule

The Power Rule is the simplest differentiation rule. It states that if you have a function of the form f(x) = x^n, where n is a constant, then the derivative is f'(x) = n*x^(n-1).

Formula: If f(x) = x^n, then f'(x) = n*x^(n-1)

For example, if f(x) = x³, then f'(x) = 3x². If f(x) = x^(1/2), then f'(x) = (1/2)x^(-1/2).

Product Rule

The Product Rule is used when you need to differentiate the product of two functions. If you have u(x) and v(x), the derivative of their product is u'v + uv'.

Formula: If f(x) = u(x)*v(x), then f'(x) = u'(x)*v(x) + u(x)*v'(x)

For example, if f(x) = x²*sin(x), then f'(x) = 2x*sin(x) + x²*cos(x).

Quotient Rule

The Quotient Rule is used when you need to differentiate a quotient of two functions. If you have u(x) and v(x), the derivative of their quotient is (u'v - uv')/v².

Formula: If f(x) = u(x)/v(x), then f'(x) = [u'(x)*v(x) - u(x)*v'(x)] / [v(x)]²

For example, if f(x) = x²/x, then f'(x) = (2x*x - x²*1)/x² = x/x² = 1/x.

Chain Rule

The Chain Rule is used for composite functions, where one function is nested inside another. If you have f(g(x)), the derivative is f'(g(x))*g'(x).

Formula: If f(x) = h(g(x)), then f'(x) = h'(g(x))*g'(x)

For example, if f(x) = sin(x²), then f'(x) = cos(x²)*2x.

Worked Examples

Example 1: Power Rule

Find the derivative of f(x) = 3x⁴ + 2x² - 5.

Solution:

Differentiate each term separately.
f'(x) = 4*3x³ + 2*2x - 0 = 12x³ + 4x.

Final answer: f'(x) = 12x³ + 4x.

Example 2: Product Rule

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

Solution:

Let u(x) = x and v(x) = e^x.
u'(x) = 1 and v'(x) = e^x.
f'(x) = u'v + uv' = 1*e^x + x*e^x = e^x(1 + x).

Final answer: f'(x) = e^x(1 + x).

Example 3: Chain Rule

Find the derivative of f(x) = cos(2x).

Solution:

Let h(u) = cos(u) and u(x) = 2x.
h'(u) = -sin(u) and u'(x) = 2.
f'(x) = h'(u)*u' = -sin(2x)*2 = -2sin(2x).

Final answer: f'(x) = -2sin(2x).

FAQ

What is the difference between differentiation and integration?

Differentiation finds the rate of change of a function, while integration finds the area under a curve or the accumulation of quantities. They are inverse processes in calculus.

When should I use the Product Rule?

You should use the Product Rule when you need to differentiate the product of two functions. It's essential for functions like x*sin(x) or x*e^x.

What is the Chain Rule used for?

The Chain Rule is used for composite functions, where one function is nested inside another. It's essential for differentiating functions like sin(x²) or e^(x³).

Can I differentiate a constant?

Yes, the derivative of any constant is zero. This is because constants do not change as the input changes.

What if I'm not sure which rule to use?

Start by identifying the structure of the function. If it's a simple power, use the Power Rule. If it's a product, use the Product Rule. If it's a quotient, use the Quotient Rule. For nested functions, use the Chain Rule.