Calculus I: Derivatives - Rules for Basic Functions and Linearity

Derivative Notation

  • The derivative of a function f(x)f(x) can be denoted as f′(x)f'(x) or ddxf(x)\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: ddx(c)=0\frac{d}{dx}(c) = 0 where cc is a constant.

  • Example: If f(x)=7π3f(x) = 7 \sqrt[3]{\pi}, then f′(x)=0f'(x) = 0 because 7π37 \sqrt[3]{\pi} is a constant number (it contains no xx variable).

  • Conceptual Understanding: Plotting a constant function (e.g., y=5y = 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., 3x23x^2), it should be handled differently (see Linearity below).

Derivative of f(x)=xf(x) = x

  • Rule: The derivative of xx with respect to xx is 11.

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

  • Conceptual Understanding: The graph of y=xy = x is a straight line with a slope of 11 (from y=mx+by = mx + b, where m=1m=1 and b=0b=0).

  • Connection to Power Rule: This can be derived using the power rule by thinking of xx as x1x^1. Applying the power rule: 1⋅x1−1=1⋅x0=1⋅1=11 \cdot x^{1-1} = 1 \cdot x^0 = 1 \cdot 1 = 1 (assuming x≠0x \neq 0).

The Power Rule

Definition

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

  • Mechanics: Take the original power nn, move it down in front of xx as a multiplier, and then subtract 11 from the original power to get the new exponent.

Applications of the Power Rule

  • Example 1: f(x)=x3f(x) = x^3

    • f′(x)=3x3−1=3x2f'(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)=1xf(x) = \frac{1}{x}

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

    • Apply power rule: f′(x)=(−1)x−1−1=−1x−2=−1x2f'(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)=1x5f(x) = \frac{1}{x^5}

    • Rewrite: f(x)=x−5f(x) = x^{-5}.

    • Apply power rule: f′(x)=(−5)x−5−1=−5x−6f'(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)=xf(x) = \sqrt{x}

    • First, rewrite as a power function: f(x)=x1/2f(x) = x^{1/2}.

    • Apply power rule: f′(x)=12x12−1=12x12−22=12x−1/2f'(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 12x\frac{1}{2\sqrt{x}}.

    • Fractional algebra (e.g., 12−1=−12\frac{1}{2} - 1 = -\frac{1}{2}) is necessary.

  • Example 5 (More Fractional Powers): f(x)=x7f(x) = \sqrt[7]{x}

    • Rewrite: f(x)=x1/7f(x) = x^{1/7}.

    • Apply power rule: f′(x)=17x17−1=17x17−77=17x−6/7f'(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)=xπf(x) = x^\pi

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

    • This is the exact answer. Do not approximate π\pi (e.g., 3.143.14) or π−1\pi - 1. Leave it in this form for mathematical exactness.

Linearity of the Derivative

Constant Multiple Rule

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

  • Concept: The constant cc