Derivatives

Derivatives

In this topic, you will learn about derivatives, which are a fundamental concept in calculus. A derivative measures how a function changes as its input changes. Understanding derivatives is essential for solving problems in various fields, such as physics and engineering. They help you understand rates of change, slopes of curves, and optimization problems.

Understanding Derivatives

Definition of a Derivative

The derivative of a function measures the rate at which the function's value changes as its input changes. It is often denoted as f'(x) or dy/dx, where y = f(x). The derivative can be thought of as the slope of the tangent line to the curve at any given point.

Basic Rules of Differentiation

There are several rules that you can use to find derivatives easily:

• Power Rule: If f(x) = x^n, then f'(x) = n * x^(n-1).
• Constant Rule: If f(x) = k (where k is a constant), then f'(x) = 0.
• Sum Rule: If f(x) = g(x) + h(x), then f'(x) = g'(x) + h'(x).
• Product Rule: If f(x) = g(x) * h(x), then f'(x) = g'(x) * h(x) + g(x) * h'(x).
• Quotient Rule: If f(x) = g(x) / h(x), then f'(x) = (g'(x) * h(x) - g(x) * h'(x)) / (h(x))^2.

Worked Example: Power Rule

Find the derivative of the function f(x) = 3x^4.

Using the power rule:

f'(x) = 4 * 3x^(4-1) = 12x^3

Thus, the derivative of f(x) = 3x^4 is f'(x) = 12x^3.

Worked Example: Product and Quotient Rules

Let y = f(x) / g(x), where f(x) = 2 + x and g(x) = x^2 + 2x + 7. Find y'.

First, find the derivatives:

f'(x) = 1

g'(x) = 2x + 2

Now, apply the quotient rule:

y' = (f' * g - f * g') / g^2

Substituting the values:

y' = (1 * (x^2 + 2x + 7) - (2 + x)(2x + 2)) / (x^2 + 2x + 7)^2

Now simplify:

y' = (x^2 + 2x + 7 - (2x^2 + 6x + 4)) / (x^2 + 2x + 7)^2

y' = (-x^2 - 4x + 3) / (x^2 + 2x + 7)^2

Implicit Differentiation

Sometimes, a function is not given explicitly in terms of y. For example, the equation x^2 + y^2 = 4 defines y implicitly. To find the derivative, we use implicit differentiation.

Worked Example: Implicit Differentiation

Find the derivative of the function x^2 + y^2 = 4.

Differentiate both sides with respect to x:

2x + 2y(dy/dx) = 0

Now, solve for dy/dx:

2y(dy/dx) = -2x

dy/dx = -x/y

Thus, the derivative of the implicit function is dy/dx = -x/y.

Summary

• The derivative measures how a function changes with respect to its input.
• Use the power, product, and quotient rules to find derivatives of functions.
• Implicit differentiation is used for functions not expressed explicitly in terms of y.

Check your understanding

1. What is the derivative of f(x) = 5x^3?
2. Use the quotient rule to find the derivative of y = (3x + 2) / (x^2 + 1).
3. Find dy/dx for the implicit function x^2 + y^2 = 9.