Lecture Note: §3 Derivatives

Last Update: 2025-10-24

§3.1 Tangents and the Derivative at a Point

• Tangents
  • The slope of the curve at \(x=c\): \(m = \lim_{h\to 0} \dfrac{f(c + h) - f(c)}{h}\)

• The derivative of a function \(f\) at a point \(c\): \(f’(c) = \lim_{h\to 0} \dfrac{f(c + h) - f(c)}{h}\)

  • \(f’(c) = \lim_{x \to c} \dfrac{f(x) - f(c)}{x - c}\)
  • means of derivative

§3.2 The Derivative as a Function

• The derivative function \(f’(x) = \lim_{h\to 0} \dfrac{f(x + h) - f(x)}{h}\)

  • The domain of \(f’\)
  • \(f’(x) = \lim_{z\to x} \dfrac{f(z) - f(x)}{z - x}\)
  • Notations: \(f’(x) = y’ = \dfrac{dy}{dx} = \dfrac{df}{dx} = \dfrac{d}{dx}f(x) = D(f)(x) = D_xf(x)\)
  • Graphing the derivative
  • When does a function not have a derivative at a point?

• Differentiable functions are continuous

§3.3 Differentiation Rule

It would be tedious if we always had to use the definition, so we develop rules for finding derivatives without having to use the definition directly.

• Derivative of some common functions
  • constant function
  • pow function (integer, general)
• New Derivatives from Old
  • constant multiple rule
  • derivative sum rule
  • product rule
  • quotient rule

§3.4 The Derivative as a Rate of Change

• Interpretation of the derivative: instantaneous rate of change
  • displacement, average velocity
• Velocity, Speed, and Acceleration

§3.5 Derivatives of Trigonometric Functions

• Derivatives of the Sine and Cosine Functions

  • \(\dfrac{d}{dx}(\sin x) = \cos x\), \(\dfrac{d}{dx}(\cos x) = -\sin x\)

• Derivatives of the Other Basic Trigonometric Functions

  • \(\dfrac{d}{dx}(\tan x) = \sec^{2} x\), \(\dfrac{d}{dx}(\cot x) = -\csc^{2} x\)

  • \(\dfrac{d}{dx}(\sec x) = \sec x \tan x\), \(\dfrac{d}{dx}(\csc x) = -\csc x \cot x\)

§3.6 The Chain Rule

We know how to differentiate both \(f\) and \(g\), so it would be useful to have a rule that tells us how to find the derivative of \(f\circ g\) in terms of the derivatives of \(f\) and \(g\).

• The Chain Rule
  • The derivative of the composite function \(f\circ g\) is the product of the derivatives of \(f\) and \(g\).
    • interpret derivatives as rates of change
  • \(f(g(x))’ = f’(g(x))\cdot g’(x)\)
  • \(\dfrac{dy}{dx} = \dfrac{dy}{du}\dfrac{du}{dx}\)
  • “Outside-Inside” Rule:
    • Differentiate the “outside” function \(f\) and evaluate this derivative at the “inside” function \(g(x)\) left alone; then multiply by the derivative of the “inside function.”
• L’Hôpital’s Rule:
  • Suppose that \(\lim_{x\to a} f (x) = \lim_{x\to a} g(x) = 0 \), that \(f\) and \(g\) are differentiable on an open interval \(I\) containing \(a\), and that \(g’(x) \neq 0\) on \(I\) if \(x\neq 0\). Then \(\lim_{x \to a}\dfrac{f(x)}{g(x)} = \lim_{x \to a}\dfrac{f’(x)}{g’(x)}\), assuming that the limit on the right side of this equation exists.

§3.7 Implicit Differentiation

• Implicit Differentiation.
  1. Differentiate both sides of the equation with respect to \(x\), treating \(y\) as a differentiable function of \(x\).
  2. Collect the terms with \(\frac{dy}{dx}\) on one side of the equation and solve for \(\frac{dy}{dx}\)
• Derivatives of Higher Order

§3.8 Related Rates

In a related rates problem, the idea is to compute the rate of change of one quantity in terms of the rate of change of another quantity (which may be more easily measured).

• The procedure is to find an equation that relates the two quantities and then use the Chain Rule to differentiate both sides with respect to time.

§3.9 Linearization and Differentials

• Linearization: linear approximation of \(f(x)\) at the point \(x=a\)
  • \(L(x) = f(a) + f’(a)(x - a)\)
  • \(f(x) \approx L(x)\) (near \(x = a\))
• Differentials
  • \(dy = f’(x) dx\)
  • \(\Delta y = f’(x)\Delta x + \varepsilon\Delta x\), where \(\varepsilon \to 0\) as \(\Delta x \to 0\).