Lecture 8: Differentiation using First Principles and Basic Rules

Differentiation Using First Principles and Limit Processes

• Introduction to Limit Processes:

  • Let $y = f(x)$ be a real-valued and continuous function.

  • The function must be defined at the point $x = x_0$.

  • The slope of the function at $x = x_0$ is defined by using a limit process.

• Geometrical Interpretation:

  • Consider the graph of $y = f(x)$ with points $P(x_0, f(x_0))$ and $Q(x_0 + h, f(x_0 + h))$.

  • The line connecting $P$ and $Q$ is the Secant Line.

  • The slope of the secant line ($m_{sec}$) is given by the formula:       $m_{sec} = \frac{f(x_0 + h) - f(x_0)}{h}$

  • In this context, $h$ represents the change in $x$ ($\Delta x$), notably $x - x_0 = h = \Delta x$.

  • As point $Q$ moves towards point $P$ along the graph, the distance $h$ becomes smaller and smaller ($h \rightarrow 0$).

  • The secant line gradually limitingly becomes the Tangent Line to the graph at point $P$.

  • The limiting value of the secant slope as $h \rightarrow 0$ is defined as the derivative of $y = f(x)$ at $x = x_0$.

Formal Definitions and Notations of the Derivative

• Notations:

  • The derivative at a point $x_0$ can be denoted by:

    • $f'(x_0)$

    • $\frac{dy}{dx}(x_0)$

    • $y'(x_0)$

    • $D_x f(x_0)$

• Mathematical Formulas:

  • $f'(x_0) = \lim_{x \rightarrow x_0} \frac{f(x) - f(x_0)}{x - x_0}$

  • Alternatively: $f'(x_0) = \lim_{h \rightarrow 0} \frac{f(x_0 + h) - f(x_0)}{h}$

• Interpretations of $f'(x_0)$:

  • (i) The slope of the tangent line to $y = f(x)$ at $x = x_0$.

  • (ii) The slope of the function at the specific point $x = x_0$.

• General Derivative:

  • If the derivative is desired at any arbitrary point $x$ within the domain of $f(x)$, the following limit process is used:       $f'(x) = \lim_{h \rightarrow 0} \frac{f(x + h) - f(x)}{h}$

Worked Examples: Limit Processes for finding Slopes and Equations

• Example 1: Quadratic Function:

  • Let $y = x^2 + 9$.

  • (i) Find $f'(2)$ using method 1:

    • $f(2) = 2^2 + 9 = 13$

    • $f'(2) = \lim_{x \rightarrow 2} \frac{(x^2 + 9) - 13}{x - 2} = \lim_{x \rightarrow 2} \frac{x^2 - 4}{x - 2}$

    • Using factorization: $\lim_{x \rightarrow 2} \frac{(x - 2)(x + 2)}{x - 2} = \lim_{x \rightarrow 2} (x + 2) = 4$

  • (ii) Find $f'(2)$ using method 2 ($h \rightarrow 0$):

    • $f(2 + h) = (2 + h)^2 + 9 = 4 + 4h + h^2 + 9 = 13 + 4h + h^2$

    • $f'(2) = \lim_{h \rightarrow 0} \frac{(13 + 4h + h^2) - 13}{h} = \lim_{h \rightarrow 0} \frac{4h + h^2}{h} = \lim_{h \rightarrow 0} (4 + h) = 4$

  • (iii) Equation of the tangent line at $x = 2$:

    • Point: $(2, 13)$. Slope: $m = 4$.

    • Using $y - y_1 = m(x - x_1)$: $y - 13 = 4(x - 2) \rightarrow y - 13 = 4x - 8 \rightarrow y = 4x + 5$.

• Example 2: Reciprocal Function:

  • Let $f(x) = \frac{1}{x}$ for $x \neq 0$.

  • Step 1: Compute $f'(x)$:

    • $f'(x) = \lim_{h \rightarrow 0} \frac{\frac{1}{x+h} - \frac{1}{x}}{h} = \lim_{h \rightarrow 0} \frac{\frac{x - (x + h)}{x(x + h)}}{h}$

    • $f'(x) = \lim_{h \rightarrow 0} \frac{-h}{h \cdot x(x + h)} = \lim_{h \rightarrow 0} \frac{-1}{x(x + h)} = -\frac{1}{x^2}$

  • Step 2: Evaluate $f'(1)$:

    • $f'(1) = -\frac{1}{1^2} = -1$

  • Step 3: Equation of tangent at $(1, 1)$:

    • Point: $(1, 1)$. Slope: $m = -1$.

    • $y - 1 = -1(x - 1) \rightarrow y = -x + 2$.

• Example 3: General Quadratic Polynomial:

  • Let $y = 2 + 3x - 3x^2$. Find $f'(2)$.

  • $f(2) = 2 + 3(2) - 3(2)^2 = 8 - 12 = -4$

  • $f(2 + h) = 2 + 3(2 + h) - 3(2 + h)^2 = 2 + 6 + 3h - 3(4 + 4h + h^2)$

  • $f(2 + h) = 8 + 3h - 12 - 12h - 3h^2 = -4 - 9h - 3h^2$

  • $f'(2) = \lim_{h \rightarrow 0} \frac{(-4 - 9h - 3h^2) - (-4)}{h} = \lim_{h \rightarrow 0} \frac{-9h - 3h^2}{h} = \lim_{h \rightarrow 0} (-9 - 3h) = -9$

• Example 4: Square Root Function:

  • Let $f(x) = \sqrt{x}$ for x > 0.

  • Find $f'(x)$ using colonization/rationalization:

    • $f'(x) = \lim_{h \rightarrow 0} \frac{\sqrt{x + h} - \sqrt{x}}{h}$

    • Multiply by the conjugate: $\frac{\sqrt{x + h} - \sqrt{x}}{h} \cdot \frac{\sqrt{x + h} + \sqrt{x}}{\sqrt{x + h} + \sqrt{x}} = \frac{(x + h) - x}{h(\sqrt{x + h} + \sqrt{x})}$

    • $f'(x) = \lim_{h \rightarrow 0} \frac{h}{h(\sqrt{x + h} + \sqrt{x})} = \lim_{h \rightarrow 0} \frac{1}{\sqrt{x + h} + \sqrt{x}} = \frac{1}{2\sqrt{x}}$

Tangent Lines and Normal Lines

• Definitions:

  • Tangent Line: A line that touches the curve at a specific point $x = x_0$.

  • Normal Line: A line drawn perpendicular to the tangent line at the same point $x = x_0$.

• Slopes:

  • Slope of tangent line at $x_0$ is $m_t = f'(x_0)$.

  • Slope of normal line at $x_0$ is the negative reciprocal: $m_n = -\frac{1}{f'(x_0)}$.

• Equations:

  • Tangent Line Equation at $(x_0, y_0)$:       $y - y_0 = f'(x_0)(x - x_0)$

  • Normal Line Equation at $(x_0, y_0)$:       $y - y_0 = -\frac{1}{f'(x_0)}(x - x_0)$, provided $f'(x_0) \neq 0$

Rules of Differentiation for Monomials and Constants

• Constant Function Rule:

  • If $f(x) = c$ (where $c$ is a constant), then $f'(x) = 0$.

  • The slope of a constant horizontal line is zero at any point.

• Power Rule for Differentiation:

  • If $f(x) = x^n$, where $n \in \mathbb{R}$, then:       $f'(x) = nx^{n-1}$

  • Derivation of the Power Rule:

    • Expand $(x + h)^n$ using the Binomial Theorem:         $(x + h)^n = \binom{n}{0}x^n + \binom{n}{1}x^{n-1}h + \binom{n}{2}x^{n-2}h^2 + ... + h^n$

    • $(x + h)^n = x^n + nx^{n-1}h + \frac{n(n-1)}{2!}x^{n-2}h^2 + ... + h^n$

    • Subtracting $x^n$ and dividing by $h$:         $\frac{(x+h)^n - x^n}{h} = nx^{n-1} + \frac{n(n-1)}{2!}x^{n-2}h + ... + h^{n-1}$

    • Taking the limit as $h \rightarrow 0$, all terms with $h$ vanish, leaving:         $f'(x) = nx^{n-1}$

• Monomial Examples:

  • If $f(x) = 5$, then $f'(x) = 0$.

  • If $f(x) = \pi$, then $f'(x) = 0$.

  • If $f(x) = -15$, then $D_x(-15) = 0$.

  • If $f(x) = x^3$, then $f'(x) = 3x^2$.

  • If $f(x) = x$, then $f'(x) = 1x^0 = 1$.

  • If $f(x) = x^{-4}$, then $f'(x) = -4x^{-5}$.

  • If $f(x) = \sqrt{x} = x^{\frac{1}{2}}$, then $f'(x) = \frac{1}{2}x^{-\frac{1}{2}}$.

General Power Rules for Differentiation

• Summary of Power Laws:

  • 1. $f(x) = x^n \Rightarrow f'(x) = nx^{n-1}$

  • 2. $f(x) = cx^n \Rightarrow f'(x) = cnx^{n-1}$

  • 3. $f(x) = x^{-n} \Rightarrow f'(x) = -nx^{-n-1}$

  • 4. $f(x) = \frac{1}{x} \Rightarrow f'(x) = -\frac{1}{x^2}$

  • 5. $f(x) = \sqrt[m]{x^n} = x^{\frac{n}{m}} \Rightarrow f'(x) = \frac{n}{m}x^{\frac{n}{m} - 1}$

• Worked Power Rule Applications:

  • Let $f(x) = \frac{3}{x^4}$. Rewrite as $3x^{-4}$. $f'(x) = 3(-4)x^{-5} = -12x^{-5}$.

  • Let $f(x) = \frac{4}{x^5}$. Rewrite as $4x^{-5}$. $f'(x) = 4(-5)x^{-6} = -20x^{-6}$.

  • Let $f(x) = \sqrt[3]{x^2}$. Rewrite as $x^{\frac{2}{3}}$. Thus for $6\sqrt[3]{x^2}$, $f'(x) = 6(\frac{2}{3})x^{-\frac{1}{3}} = 4x^{-\frac{1}{3}}$.

Differentiation of Polynomials

• Formula:

  • Let $f(x) = a_n x^n + ... + a_2 x^2 + a_1 x + a_0$.

  • Then $f'(x) = a_n(nx^{n-1}) + ... + a_2(2x) + a_1(1) + 0$.

• Examples:

  • (i) $f(b) = b^3 + 4b + 8 \Rightarrow f'(b) = 3b^2 + 4$.

  • (ii) $f(x) = 5x^3 + 4x^2 - 3x + 10 \Rightarrow f'(x) = 15x^2 + 8x - 3$.

  • (iii) $f(b) = 6b^3 - 4b^2 + 21 \Rightarrow f'(b) = 18b^2 - 8b$.

Differentiation of Exponential Functions

• Basic Rules:

  • If $y = e^x$, then $\frac{dy}{dx} = e^x$.

  • If $y = e^{-x}$, then $\frac{dy}{dx} = -e^{-x}$.

  • $e \approx 2.71828$ and $\ln(e) = 1$.

• General Exponential Rule:

  • If $y = a^x$ (where $a$ is a positive constant), then:       $y' = a^x \ln(a)$

  • Example: $y = 5^x \Rightarrow y' = 5^x \ln(5)$.

• Generalized Rules (Chain Rule Extension):

  • (i) If $y = a^{u(x)}$, then $y' = a^{u(x)} \cdot u'(x) \cdot \ln(a)$.

  • (ii) If $y = e^{u(x)}$, then $y' = e^{u(x)} \cdot u'(x)$.

  • (iii) If $y = [u(x)]^{v(x)}$, then:       $f'(x) = [u(x)]^{v(x)} [v'(x) \ln u(x) + v(x) \frac{u'(x)}{u(x)}]$

• Worked Examples:

  • 1. $f(t) = e^{t^2}$. Let $u(t) = t^2$, $u'(t) = 2t$. Result: $f'(t) = 2te^{t^2}$.

  • 2. $f(t) = 5^{t^3}$. $u(t) = t^3$, $u'(t) = 3t^2$. Result: $f'(t) = 3t^2 \cdot 5^{t^3} \cdot \ln(5)$.

  • 3. $f(x) = x^{x^2}$. $u(x) = x$, $v(x) = x^2$. Result: $f'(x) = x^{x^2} [2x \ln(x) + x^2 \cdot \frac{1}{x}] = x^{x^2} [2x \ln(x) + x]$.

Differentiation of Logarithmic Functions

• Basic Forms (Base $e$):

  • If $y = \ln(x)$, then $y' = \frac{1}{x}$.

• Arbitrary Base ($base \, a$):

  • If $f(x) = \log_a(x)$, where $a \neq 1$, then $f'(x) = \frac{1}{x \ln(a)}$.

• General Forms (Chain Rule Extension):

  • (i) If $y = \ln u(x)$, then $y' = \frac{u'(x)}{u(x)}$.

  • (ii) If $y = \log_a u(x)$, then $y' = \frac{u'(x)}{u(x) \ln(a)}$.

• Worked Examples:

  • 1. $f(x) = \log_2(x^2 + 2x + 1)$. $u(x) = x^2 + 2x + 1$, $u'(x) = 2x + 2$.       Result: $f'(x) = \frac{2x + 2}{(x^2 + 2x + 1) \ln(2)}$.

  • 2. $f(x) = \ln(x^2 + 2x + 1)$.       Result: $f'(x) = \frac{2x + 2}{x^2 + 2x + 1}$.

Existence Conditions for the Derivative

• Main Conditions for $f'(x_0)$ to exist:

  • (i) $x_0$ must be in the domain of $f(x)$. (Function defined at $x_0$).

  • (ii) $f(x)$ is continuous at $x = x_0$. $\lim_{x \rightarrow x_0^-} f(x) = \lim_{x \rightarrow x_0^+} f(x) = f(x_0)$.

  • (iii) The graph of $y = f(x)$ has no sharp edges, kinks, or cusps at $x = x_0$.

  • (iv) There is No vertical tangent at $x = x_0$.

  • (v) There is No vertical asymptote at $x = x_0$.

• Examples where the derivative does not exist (DNE):

  • At holes/discontinuities where $x_0$ is not defined.

  • At points of jump discontinuity.

  • At a cusp or sharp corner/edge.

  • At points with a vertical tangent.

  • At points with a vertical asymptote.

Differentiation of Trigonometric Functions

• Basic Formulas:

  • $\frac{d}{dx} \sin(x) = \cos(x)$

  • $\frac{d}{dx} \cos(x) = -\sin(x)$

  • $\frac{d}{dx} \tan(x) = \sec^2(x)$

  • $\frac{d}{dx} \sec(x) = \sec(x) \tan(x)$

• Generalized Formulas (Chain Rule):

  • (i) $\frac{d}{dx} \sin(u(x)) = \cos(u(x)) \cdot u'(x)$

  • (ii) $\frac{d}{dx} \cos(u(x)) = -\sin(u(x)) \cdot u'(x)$

  • (iii) $\frac{d}{dx} \tan(u(x)) = \sec^2(u(x)) \cdot u'(x)$

  • (iv) $\frac{d}{dx} \sec(u(x)) = \sec(u(x)) \tan(u(x)) \cdot u'(x)$

• Worked Examples:

  • 1. $y = e^{\sin(x)}$. Here $u(x) = \sin(x)$, $u'(x) = \cos(x)$. Result: $y' = \cos(x)e^{\sin(x)}$.

  • 2. $y = \cos(2x + 1)$. Here $u(x) = 2x + 1$, $u'(x) = 2$. Result: $y' = -2 \sin(2x + 1)$.

  • 3. $y = \tan(3x^2 + 1)$. Here $u(x) = 3x^2 + 1$, $u'(x) = 6x$. Result: $y' = 6x \sec^2(3x^2 + 1)$.

Higher Order Derivatives

• Notations:

  • First Order: $y' = \frac{dy}{dx} = f'(x) = D_x f(x)$.

  • Second Order: $y'' = \frac{d^2y}{dx^2} = f''(x) = D_x^2 f(x)$.

  • Third Order: $y''' = \frac{d^3y}{dx^3} = f'''(x) = D_x^3 f(x)$.

  • Fourth Order: $y^{(4)} = \frac{d^4y}{dx^4} = f^{(4)}(x) = D_x^4 f(x)$.

  • n-th Order: $y^{(n)} = \frac{d^ny}{dx^n} = f^{(n)}(x) = D_x^n f(x)$.

• Definition: If a function is infinitely differentiable, all orders of differentiation exist for all $n \in \mathbb{N}$.

More Rules: Sum and Difference Rules

• Sum Rule:

  • $\frac{d}{dx} [f(x) + g(x)] = f'(x) + g'(x)$

  • Example: $F(x) = e^{2x} + \sin(3x)$. $F'(x) = 2e^{2x} + 3 \cos(3x)$.

• Difference Rule:

  • $\frac{d}{dx} [f(x) - g(x)] = f'(x) - g'(x)$

  • Example: $F(x) = 4\sqrt{x^5} - e^{x^2+1} = 4x^{\frac{5}{2}} - e^{x^2+1}$.       $F'(x) = 4(\frac{5}{2})x^{\frac{3}{2}} - 2x e^{x^2+1} = 10x^{\frac{3}{2}} - 2x e^{x^2+1}$.

• Comprehensive Example with Higher Order:

  • Let $y(v) = ae^v + \frac{c}{v^2} = ae^v + cv^{-2}$. (where $a, c$ are constants).

  • First Derivative: $y'(v) = ae^v - 2cv^{-3}$.

  • Second Derivative: $y''(v) = \frac{d}{dv} (ae^v - 2cv^{-3}) = ae^v + 6cv^{-4}$.

  • Third Derivative: $y'''(v) = \frac{d}{dv} (ae^v + 6cv^{-4}) = ae^v - 24cv^{-5}$.