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3.3 Differentiation Rules

3 min read•june 24, 2024

Differentiation rules are the building blocks of calculus, helping us find rates of change in functions. These rules, from basic to complex, give us tools to tackle various mathematical challenges and understand how quantities change over time.

By mastering these rules, we can analyze everything from simple linear functions to complex polynomials and trigonometric expressions. This knowledge forms the foundation for solving real-world problems in physics, economics, and engineering.

Differentiation Rules

Basic differentiation rules

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  • High School Calculus/Techniques of Differentiation - Wikibooks, open books for an open world View original

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  • : The derivative of a constant is always 0
    • Formula: ddx(c)=0\frac{d}{dx}(c) = 0, where cc is a constant (5, -2.7, π)
  • : Multiplying a function by a constant multiplies the derivative by the same constant
    • Formula: ddx(cf(x))=cddx(f(x))\frac{d}{dx}(cf(x)) = c\frac{d}{dx}(f(x)), where cc is a constant (3, -1/2, e)
  • : The derivative of a variable raised to a power is the power multiplied by the variable raised to one less than the original power
    • Formula: ddx(xn)=nxn−1\frac{d}{dx}(x^n) = nx^{n-1}, where nn is a constant (2, 1/3, -4)
    • Examples: ddx(x3)=3x2\frac{d}{dx}(x^3) = 3x^2, ddx(x−1/2)=−12x−3/2\frac{d}{dx}(x^{-1/2}) = -\frac{1}{2}x^{-3/2}

Sum and difference rules

  • : The derivative of a sum of functions is the sum of their derivatives
    • Formula: ddx(f(x)+g(x))=ddx(f(x))+ddx(g(x))\frac{d}{dx}(f(x) + g(x)) = \frac{d}{dx}(f(x)) + \frac{d}{dx}(g(x))
    • Example: ddx(x2+sin⁡(x))=2x+cos⁡(x)\frac{d}{dx}(x^2 + \sin(x)) = 2x + \cos(x)
  • : The derivative of a difference of functions is the difference of their derivatives
    • Formula: ddx(f(x)−g(x))=ddx(f(x))−ddx(g(x))\frac{d}{dx}(f(x) - g(x)) = \frac{d}{dx}(f(x)) - \frac{d}{dx}(g(x))
    • Example: ddx(ex−ln⁡(x))=ex−1x\frac{d}{dx}(e^x - \ln(x)) = e^x - \frac{1}{x}

Product rule for derivatives

  • : The derivative of a product of functions is the first function multiplied by the derivative of the second function, plus the second function multiplied by the derivative of the first function
    • Formula: ddx(f(x)g(x))=f(x)ddx(g(x))+g(x)ddx(f(x))\frac{d}{dx}(f(x)g(x)) = f(x)\frac{d}{dx}(g(x)) + g(x)\frac{d}{dx}(f(x))
    • Example: ddx(x2sin⁡(x))=x2cos⁡(x)+2xsin⁡(x)\frac{d}{dx}(x^2\sin(x)) = x^2\cos(x) + 2x\sin(x)
  • The product rule is useful when differentiating functions that are multiplied together (polynomial × trigonometric, exponential × logarithmic)

Quotient rule for derivatives

  • : The derivative of a quotient of functions is the denominator multiplied by the derivative of the numerator, minus the numerator multiplied by the derivative of the denominator, all divided by the square of the denominator
    • Formula: ddx(f(x)g(x))=g(x)ddx(f(x))−f(x)ddx(g(x))[g(x)]2\frac{d}{dx}(\frac{f(x)}{g(x)}) = \frac{g(x)\frac{d}{dx}(f(x)) - f(x)\frac{d}{dx}(g(x))}{[g(x)]^2}
    • Example: ddx(x2ex)=ex(2x)−x2(ex)(ex)2=2xex−x2exe2x\frac{d}{dx}(\frac{x^2}{e^x}) = \frac{e^x(2x) - x^2(e^x)}{(e^x)^2} = \frac{2xe^x - x^2e^x}{e^{2x}}
  • The quotient rule is used when differentiating functions that are divided by each other (, trigonometric ratios)

Differentiation of negative exponents

  • Negative Exponent Rule: The derivative of a function with a negative exponent can be found by applying the power rule and the constant multiple rule
    • Formula: ddx(x−n)=−nx−n−1\frac{d}{dx}(x^{-n}) = -nx^{-n-1}, where nn is a positive constant (1, 2, 1/2)
    • This rule follows from the power rule and the constant multiple rule: ddx(x−n)=ddx(1xn)=−1xn+1⋅ddx(xn)=−nx−n−1\frac{d}{dx}(x^{-n}) = \frac{d}{dx}(\frac{1}{x^n}) = -\frac{1}{x^{n+1}} \cdot \frac{d}{dx}(x^n) = -nx^{-n-1}
    • Example: ddx(x−3)=−3x−4\frac{d}{dx}(x^{-3}) = -3x^{-4}, ddx(1x)=−12x−3/2\frac{d}{dx}(\frac{1}{\sqrt{x}}) = -\frac{1}{2}x^{-3/2}

Complex function differentiation techniques

  • When differentiating complex functions, identify the individual components and apply the appropriate differentiation rules to each component
    1. Break down the function into simpler parts using sum, difference, product, and quotient rules
    2. Apply the power rule, constant multiple rule, and negative exponent rule to each part as needed
    3. Combine the derivatives of the individual parts according to the rule used to break down the function
  • Polynomial functions can be differentiated by applying the sum rule and the power rule to each term
    • Example: ddx(3x4−2x3+5x−1)=12x3−6x2+5\frac{d}{dx}(3x^4 - 2x^3 + 5x - 1) = 12x^3 - 6x^2 + 5
  • Rational functions can be differentiated by applying the quotient rule, and then using other differentiation rules to simplify the resulting expression
    • Example: ddx(x2+3x2x−1)=(2x−1)(2x+3)−(x2+3x)(2)(2x−1)2=4x2−2x+3−2x2−6x4x2−4x+1=2x2−8x+34x2−4x+1\frac{d}{dx}(\frac{x^2 + 3x}{2x - 1}) = \frac{(2x - 1)(2x + 3) - (x^2 + 3x)(2)}{(2x - 1)^2} = \frac{4x^2 - 2x + 3 - 2x^2 - 6x}{4x^2 - 4x + 1} = \frac{2x^2 - 8x + 3}{4x^2 - 4x + 1}
  • Implicit differentiation can be used when a function is not explicitly defined in terms of x

Advanced concepts in differentiation

  • The concept of a is fundamental to understanding derivatives, as derivatives are defined as the limit of a difference quotient
  • is closely related to differentiability, as a function must be continuous at a point to be differentiable there
  • Higher-order derivatives involve taking the derivative of a derivative, allowing for more complex analysis of function behavior

Key Terms to Review (14)

Constant multiple law for limits: The Constant Multiple Law for limits states that the limit of a constant multiplied by a function is equal to the constant multiplied by the limit of the function. Mathematically, if $\lim_{{x \to c}} f(x) = L$, then $\lim_{{x \to c}} [k \cdot f(x)] = k \cdot L$ where $k$ is a constant.
Constant Multiple Rule: The Constant Multiple Rule states that the derivative of a constant multiplied by a function is the constant multiplied by the derivative of the function. Mathematically, if $c$ is a constant and $f(x)$ is a differentiable function, then $(cf(x))' = c f'(x)$.
Constant rule: The constant rule states that the derivative of a constant function is zero. If $c$ is a constant, then $\frac{d}{dx}c = 0$.
Continuity: Continuity is a fundamental concept in calculus that describes the smoothness and uninterrupted nature of a function. It is a crucial property that allows for the application of calculus techniques and the study of limits, derivatives, and integrals.
Continuity over an interval: Continuity over an interval means that a function is continuous at every point within a given interval. This implies that the function has no breaks, jumps, or holes in that interval.
Difference Rule: The Difference Rule states that the derivative of a difference of two functions is the difference of their derivatives. Mathematically, if $f(x)$ and $g(x)$ are differentiable, then $(f-g)' = f' - g'$.
Limit: In mathematics, the limit of a function is a fundamental concept that describes the behavior of a function as its input approaches a particular value. It is a crucial notion that underpins the foundations of calculus and serves as a building block for understanding more advanced topics in the field.
Point-slope equation: The point-slope equation of a line is given by $y - y_1 = m(x - x_1)$, where $(x_1, y_1)$ is a point on the line and $m$ is the slope. It is useful for writing the equation of a line when you know one point and the slope.
Power rule: The power rule is a basic differentiation rule used to find the derivative of a function of the form $f(x) = x^n$. It states that if $f(x) = x^n$, then $f'(x) = nx^{n-1}$.
Product rule: The product rule is a differentiation rule used to find the derivative of the product of two functions. It states that if $u(x)$ and $v(x)$ are differentiable functions, then the derivative of their product is given by $(uv)' = u'v + uv'$.
Quotient rule: The quotient rule is a formula for finding the derivative of the ratio of two differentiable functions. It states that if $u(x)$ and $v(x)$ are functions, then $(\frac{u}{v})' = \frac{u'v - uv'}{v^2}$.
Rational Functions: A rational function is a function that can be expressed as the ratio of two polynomial functions. It is a mathematical expression that can be used to model a wide range of real-world phenomena, from population growth to the behavior of electrical circuits.
Slope: Slope is a measure of the steepness or incline of a line or curve, representing the rate of change between two points. It is a fundamental concept in calculus that underpins the understanding of functions, rates of change, and the shape of graphs.
Sum Rule: The Sum Rule states that the derivative of the sum of two functions is equal to the sum of their derivatives. Mathematically, if $f(x)$ and $g(x)$ are differentiable functions, then $(f+g)'(x) = f'(x) + g'(x)$.
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Glossary
3.4 Derivatives as Rates of Change