Calculus I: Derivatives - Rules for Basic Functions and Linearity

Derivative Notation

• The derivative of a function f(x) can be denoted as f'(x) or \frac{d}{dx}f(x).

• Both notations mean the same thing and will be used interchangeably.

Derivatives of Basic Functions

Derivatives of Constants

• Rule: The derivative of any constant value is always zero.

  • Mathematically: \frac{d}{dx}(c) = 0 where c is a constant.

• Example: If f(x) = 7 \sqrt[3]{\pi}, then f'(x) = 0 because 7 \sqrt[3]{\pi} is a constant number (it contains no x variable).

• Conceptual Understanding: Plotting a constant function (e.g., y = 5) results in a horizontal line, and the slope of a horizontal line is zero.

• Important Distinction: This rule applies when the entire function is a constant. If a constant is part of a larger function (e.g., 3x^2), it should be handled differently (see Linearity below).

Derivative of f(x) = x

• Rule: The derivative of x with respect to x is 1.

  • Mathematically: \frac{d}{dx}(x) = 1 or x' = 1.

• Conceptual Understanding: The graph of y = x is a straight line with a slope of 1 (from y = mx + b, where m=1 and b=0).

• Connection to Power Rule: This can be derived using the power rule by thinking of x as x^1. Applying the power rule: 1 \cdot x^{1-1} = 1 \cdot x^0 = 1 \cdot 1 = 1 (assuming x \neq 0).

The Power Rule

Definition

• Rule: If f(x) = x^n, then its derivative is f'(x) = n x^{n-1}.

• Mechanics: Take the original power n, move it down in front of x as a multiplier, and then subtract 1 from the original power to get the new exponent.

Applications of the Power Rule

• Example 1: f(x) = x^3

  • f'(x) = 3x^{3-1} = 3x^2.

  • Historically, this was proven through a lengthy limit definition process, but the power rule provides a quick method.

• Example 2 (Negative Powers): f(x) = \frac{1}{x}

  • First, rewrite as a power function: f(x) = x^{-1}.

  • Apply power rule: f'(x) = (-1)x^{-1-1} = -1x^{-2} = -\frac{1}{x^2}.

  • This matches results from the limit definition.

• Example 3 (More Negative Powers): f(x) = \frac{1}{x^5}

  • Rewrite: f(x) = x^{-5}.

  • Apply power rule: f'(x) = (-5)x^{-5-1} = -5x^{-6}.

  • Note on Negative Exponents: In calculus, negative powers are often preferred for answers, especially if further derivatives are to be taken, as they are already in the correct form for the power rule.

• Example 4 (Fractional Powers - Roots): f(x) = \sqrt{x}

  • First, rewrite as a power function: f(x) = x^{1/2}.

  • Apply power rule: f'(x) = \frac{1}{2}x^{\frac{1}{2}-1} = \frac{1}{2}x^{\frac{1}{2}-\frac{2}{2}} = \frac{1}{2}x^{-1/2}.

  • This can also be written as \frac{1}{2\sqrt{x}}.

  • Fractional algebra (e.g., \frac{1}{2} - 1 = -\frac{1}{2}) is necessary.

• Example 5 (More Fractional Powers): f(x) = \sqrt[7]{x}

  • Rewrite: f(x) = x^{1/7}.

  • Apply power rule: f'(x) = \frac{1}{7}x^{\frac{1}{7}-1} = \frac{1}{7}x^{\frac{1}{7}-\frac{7}{7}} = \frac{1}{7}x^{-6/7}.

• Example 6 (Irrational Exponents): f(x) = x^\pi

  • Apply power rule directly: f'(x) = \pi x^{\pi-1}.

  • This is the exact answer. Do not approximate \pi (e.g., 3.14) or \pi - 1. Leave it in this form for mathematical exactness.

Linearity of the Derivative

Constant Multiple Rule

• Rule: If c is a constant and f(x) is a differentiable function, then \frac{d}{dx}(c \cdot f(x)) = c \cdot \frac{d}{dx}(f(x)).

• Concept: The constant c