Differentiation is the key to understanding how functions change. It's all about finding slopes and rates of change at specific points. This concept is crucial for analyzing function behavior and solving real-world problems.

The rules of differentiation give us shortcuts to find derivatives quickly. From basic power rules to more complex chain rules, these tools help us tackle a wide range of functions and make calculus much more manageable.

Differentiation: Definition and Role

Concept and Definition

• Differentiation is a fundamental concept in mathematical analysis that involves finding the rate of change or slope of a function at any given point
• The derivative of a function $f(x)$ at a point $x$ is defined as the limit of the difference quotient $\frac{f(x + h) - f(x)}{h}$ as $h$ approaches $0$, provided the limit exists
  • This limit represents the instantaneous rate of change of the function at the point $x$
  • The derivative is denoted by $f'(x)$ or $\frac{d}{dx}f(x)$

Significance and Applications

• Derivatives play a crucial role in determining the behavior of functions
  • Increasing or decreasing intervals ($f'(x) > 0$ indicates increasing, $f'(x) < 0$ indicates decreasing)
  • Local extrema (critical points where $f'(x) = 0$ or $f'(x)$ does not exist)
  • Concavity ($f''(x) > 0$ indicates concave up, $f''(x) < 0$ indicates concave down)
• Differentiation is used in various applications
  • Optimization problems (finding maximum or minimum values)
  • Rate of change calculations (velocity, acceleration, marginal cost, etc.)
  • Approximation of functions using Taylor series (polynomial approximations)
  • Solving differential equations (equations involving derivatives)

Differentiation: Basic Rules and Applications

Constant and Power Rules

• The constant rule states that the derivative of a constant function is always $0$
  • If $f(x) = c$, where $c$ is a constant, then $f'(x) = 0$
  • Example: If $f(x) = 5$, then $f'(x) = 0$
• The power rule allows for the differentiation of functions of the form $x^n$, where the derivative is $nx^{n-1}$
  • If $f(x) = x^n$, then $f'(x) = nx^{n-1}$
  • Example: If $f(x) = x^3$, then $f'(x) = 3x^2$

Product and Quotient Rules

• The product rule states that the derivative of the product of two functions $u(x)$ and $v(x)$ is $u'(x)v(x) + u(x)v'(x)$
  • If $f(x) = u(x)v(x)$, then $f'(x) = u'(x)v(x) + u(x)v'(x)$
  • Example: If $f(x) = (x^2 + 1)(x - 2)$, then $f'(x) = (2x)(x - 2) + (x^2 + 1)(1)$
• The quotient rule is used to find the derivative of the quotient of two functions $u(x)$ and $v(x)$, given by $\frac{v(x)u'(x) - u(x)v'(x)}{[v(x)]^2}$, provided $v(x) \neq 0$
  • If $f(x) = \frac{u(x)}{v(x)}$, then $f'(x) = \frac{v(x)u'(x) - u(x)v'(x)}{[v(x)]^2}$
  • Example: If $f(x) = \frac{x^2 + 1}{x - 2}$, then $f'(x) = \frac{(x - 2)(2x) - (x^2 + 1)(1)}{(x - 2)^2}$

Chain Rule

• The chain rule is applied when differentiating composite functions, where the derivative of $f(g(x))$ is $f'(g(x))g'(x)$
  • If $f(x) = h(g(x))$, then $f'(x) = h'(g(x))g'(x)$
  • Example: If $f(x) = (x^2 + 1)^3$, then $f'(x) = 3(x^2 + 1)^2 \cdot 2x$
  • The chain rule can be extended to multiple compositions, such as $f(g(h(x)))$

Derivative Calculation: Definition and Rules

Using the Definition

• To find the derivative of a function using the definition, evaluate the limit of the difference quotient $\frac{f(x + h) - f(x)}{h}$ as $h$ approaches $0$
  • Substitute $x + h$ for $x$ in the function, subtract the original function, divide by $h$, and find the limit as $h$ approaches $0$
  • Example: Find the derivative of $f(x) = x^2$ using the definition
    1. $f(x + h) = (x + h)^2 = x^2 + 2xh + h^2$
    2. $\frac{f(x + h) - f(x)}{h} = \frac{(x^2 + 2xh + h^2) - x^2}{h} = \frac{2xh + h^2}{h} = 2x + h$
    3. $\lim_{h \to 0} (2x + h) = 2x$, so $f'(x) = 2x$

Applying Differentiation Rules

• When using the power rule, decrease the exponent by $1$ and multiply by the original exponent
  • Example: If $f(x) = x^5$, then $f'(x) = 5x^4$
• To apply the product rule, multiply the first function by the derivative of the second, then add the product of the second function and the derivative of the first
  • Example: If $f(x) = (x^3 + 2)(x - 1)$, then $f'(x) = (3x^2)(x - 1) + (x^3 + 2)(1)$
• When using the quotient rule, subtract the product of the numerator and the derivative of the denominator from the product of the denominator and the derivative of the numerator, then divide by the square of the denominator
  • Example: If $f(x) = \frac{x^2 - 1}{x + 2}$, then $f'(x) = \frac{(x + 2)(2x) - (x^2 - 1)(1)}{(x + 2)^2}$
• To differentiate composite functions using the chain rule, first differentiate the outer function, then multiply by the derivative of the inner function
  • Example: If $f(x) = \sin(x^2)$, then $f'(x) = \cos(x^2) \cdot 2x$

Derivative: Geometric Interpretation as Slope

Tangent Line and Slope

• The derivative of a function at a point represents the slope of the tangent line to the function's graph at that point
  • A positive derivative indicates an increasing function (tangent line has positive slope)
  • A negative derivative indicates a decreasing function (tangent line has negative slope)
  • A derivative of $0$ implies a horizontal tangent line, which may occur at local extrema or inflection points
• The equation of the tangent line to a function $f(x)$ at a point $(a, f(a))$ is given by $y - f(a) = f'(a)(x - a)$
  • The point $(a, f(a))$ is the point of tangency
  • The slope of the tangent line is equal to the derivative $f'(a)$
  • Example: Find the equation of the tangent line to $f(x) = x^2$ at the point $(2, 4)$
    1. $f'(x) = 2x$, so $f'(2) = 4$
    2. The equation of the tangent line is $y - 4 = 4(x - 2)$ or $y = 4x - 4$

Normal Line

• The normal line to a function's graph at a point is perpendicular to the tangent line and has a slope equal to the negative reciprocal of the derivative at that point
  • If the slope of the tangent line is $m$, the slope of the normal line is $-\frac{1}{m}$
  • The equation of the normal line can be found using the point-slope form, with the slope being the negative reciprocal of the derivative
  • Example: Find the equation of the normal line to $f(x) = x^2$ at the point $(2, 4)$
    1. The slope of the tangent line is $f'(2) = 4$
    2. The slope of the normal line is $-\frac{1}{4}$
    3. The equation of the normal line is $y - 4 = -\frac{1}{4}(x - 2)$ or $y = -\frac{1}{4}x + \frac{9}{2}$