In this section, we will cover the objectives of finding the derivatives of constant functions and the form f(x) = x, as well as finding derivatives of sums, differences, products, and quotients. The derivative of a constant, represented as [c], is always 0, where c is a constant. We will also discuss the power rule and the constant multiple rule, and provide examples for better understanding.
Derivatives of Polynomials and Exponential Functions Formula
The power rule states that if n is an integer, then the derivative of a function f(x) = x^n is f'(x) = nx^(n-1). The constant multiple rule specifies that if c is a constant and f is a differentiable function, then the derivative of cf(x) is c times the derivative of f(x). We will also delve into the sum and difference rules for finding the derivatives of functions.
Examples of Derivatives of Polynomials and Exponential Functions
Example 1:
Given f(x) = x^15,
The derivative f'(x) = 15x^14.
Example 2:
Given f(x) = √x,
The derivative f'(x) = x^(-1/2).
Product and Quotient Rules
Moving on, we will explore the product and quotient rules for finding the derivatives of functions. The product rule states that if f and g are both differentiable functions, then the derivative of f(x) * g(x) is f(x) * g'(x) + g(x) * f'(x). Similarly, the quotient rule for finding the derivative of f(x) / g(x) is [f'(x)g(x) - f(x)g'(x)] / [g(x)]^2.
Product and Quotient Rule Examples
Example of Product Rule:
Given f(x) = (x^2 + x) (3x + 1),
The derivative f'(x) = (x^2 + x)(3) + (3x + 1)(2x + 1).
Example of Quotient Rule:
Given f(x) = 1/(x^2 + 5x),
The derivative f'(x) = [3x(10x + e^2) + (5x^2 + e^2)] / [x^2 + 5x]^2.
Rates of Change in the Natural and Social Sciences
In this portion, we will apply the concepts learned to real-world examples. For instance, we will look at the rate of change of a particle's motion and explore scenarios such as finding the particle's velocity, determining when it is at rest, and calculating total distance traveled during a specific time period.
Example of Rates of Change
Example 1: A particle moves according to a law of motion s(t) = t^4 - 4t + 1. We will find the particle's velocity at a given time, determine when the particle is at rest, and identify when it is moving in the positive direction.
Example 2: If a ball is thrown vertically upward with a velocity of 80 ft/sec, then its height after t seconds is given by s = 80t - 16t^2. We will calculate the maximum height reached by the ball.
These concepts and examples will help in understanding the practical applications of derivatives in various fields, including physics and engineering.
By covering derivatives of polynomials and exponential functions, product and quotient rules, and practical examples of rates of change, we aim to provide a comprehensive understanding of these fundamental calculus concepts. Whether you are studying for an AP Calculus AB test or seeking to enhance your calculus skills, these concepts are crucial for mastery.
