Calculus Study Notes: The Chain Rule, Logarithmic Derivatives, and General Power Rule

The Chain Rule for Composite Functions

• Definition of Composition and Chain Rule Formula:

  • When evaluating the derivative of a composite function $g(f(x))$, where an inner function $f(x)$ is nested inside an outer function $g(x)$, the derivative is obtained by taking the derivative of the outer function evaluated at the inner function, and multiplying it by the derivative of the inner function:         $\frac{d}{dx}[g(f(x))] = g'(f(x)) \times f'(x)$

• Derivation of Square Root Derivative via Power Rule:

  • The square root function can be expressed as a fractional exponent:         $\sqrt{x} = x^{\frac{1}{2}}$

  • Applying the power rule yields:         $\frac{d}{dx}[x^{\frac{1}{2}}] = \frac{1}{2}x^{\frac{1}{2} - 1} = \frac{1}{2}x^{-\frac{1}{2}} = \frac{1}{2\sqrt{x}}$

• Worked Example 1: Square Root of a Polynomial:

  • Function:         $y = \sqrt{x^2 + x^3 + 2}$

  • Identification of Components:

    • Outer function: $g(u) = \sqrt{u}$ with derivative $g'(u) = \frac{1}{2\sqrt{u}}$

    • Inner function: $f(x) = x^2 + x^3 + 2$

  • Step-by-Step Chain Rule Solution:

    1. Differentiate outer function while retaining the inner function:             $g'(f(x)) = \frac{1}{2\sqrt{x^2 + x^3 + 2}}$

    2. Differentiate inner function:             $f'(x) = \frac{d}{dx}[x^2 + x^3 + 2] = 2x + 3x^2$

    3. Multiply the outer derivative by the inner derivative:             $\frac{dy}{dx} = \frac{1}{2\sqrt{x^2 + x^3 + 2}} \times (2x + 3x^2)$

  • Alternative Power Representation Solution:

    • Rewrite function:             $y = (x^2 + x^3 + 2)^{\frac{1}{2}}$

    • Differentiate using the general power rule:             $\frac{dy}{dx} = \frac{1}{2}(x^2 + x^3 + 2)^{-\frac{1}{2}} \times (2x + 3x^2)$

• Worked Example 2: Trigonometric Function with Nested Square Root:

  • Function:         $f(x) = \sin(x + \sqrt{x})$

  • Identification of Components:

    • Outer function: $\sin(u)$ with derivative $\cos(u)$

    • Inner function: $x + \sqrt{x}$

  • Step-by-Step Solution:

    1. Differentiate outer function while leaving inner function unchanged:             $\cos(x + \sqrt{x})$

    2. Differentiate inner function:             $\frac{d}{dx}[x + \sqrt{x}] = 1 + \frac{1}{2\sqrt{x}}$

    3. Combine via multiplication:             $f'(x) = \cos(x + \sqrt{x}) \times \left(1 + \frac{1}{2\sqrt{x}}\right)$

    • Note: The trigonometric expression $\cos(x + \sqrt{x})$ is evaluated separately and then multiplied by the full derivative of the inner term $\left(1 + \frac{1}{2\sqrt{x}}\right)$.

Derivatives of Logarithmic Functions

• Inverse Relationship Between Exponential and Logarithm Functions:

  • The exponential function $e^x$ and the natural logarithm function $\ln(x)$ are inverse functions of each other that cancel each other out when composed:         $e^{\ln(x)} = x \quad \text{for } x > 0$         $\ln(e^x) = x \quad \text{for all } x$

• Derivation of the Derivative of Natural Logarithm:

  • To find $\frac{d}{dx}[\ln(x)]$, begin with the inverse identity:         $e^{\ln(x)} = x$

  • Differentiate both sides with respect to $x$ using implicit differentiation and the chain rule:         $\frac{d}{dx}[e^{\ln(x)}] = \frac{d}{dx}[x]$

  • Apply chain rule to left side (outer function $e^u$, inner function $\ln(x)$):         $e^{\ln(x)} \times \frac{d}{dx}[\ln(x)] = 1$

  • Solve algebraically for $\frac{d}{dx}[\ln(x)]$:         $\frac{d}{dx}[\ln(x)] = \frac{1}{e^{\ln(x)}}$

  • Substitute $e^{\ln(x)} = x$ back into the denominator:         $\frac{d}{dx}[\ln(x)] = \frac{1}{x}$

Differentiation of Exponential Functions with Arbitrary Bases

• Algebraic Transformation to Base $e$:

  • For any positive constant base $a > 0$, express $a$ as an exponential function with base $e$:         $a = e^{\ln(a)}$

  • Rewrite $a^x$ using rules of exponents:         $a^x = (e^{\ln(a)})^x = e^{x\ln(a)}$

• Derivation of Derivative Formula for $a^x$:

  • Differentiate $e^{x\ln(a)}$ using the chain rule:         $\frac{d}{dx}[a^x] = \frac{d}{dx}[e^{x\ln(a)}]$

  • Outer function is $e^u$; inner function is $x\ln(a)$ (where $\ln(a)$ is a constant):         $\frac{d}{dx}[x\ln(a)] = \ln(a)$

  • Apply the chain rule:         $\frac{d}{dx}[e^{x\ln(a)}] = e^{x\ln(a)} \times \frac{d}{dx}[x\ln(a)] = e^{x\ln(a)} \times \ln(a)$

  • Substitute $e^{x\ln(a)} = a^x$ back into the equation:         $\frac{d}{dx}[a^x] = a^x \ln(a)$

• Special Case $a = e$:

  • Substituting $a = e$ yields:         $\frac{d}{dx}[e^x] = e^x \ln(e) = e^x \times 1 = e^x$

• Application to Doubling Functions:

  • For a function modeling a doubling process where $y = 2^x$:         $\frac{d}{dx}[2^x] = 2^x \ln(2)$

The General Power Rule for All Real Exponents

• Derivation for Arbitrary Real Exponents:

  • Let $r$ be any real number, and consider $y = x^r$.

  • Rewrite $x$ using the natural logarithm and exponential base $e$:         $x^r = (e^{\ln(x)})^r = e^{r\ln(x)}$

  • Differentiate both sides using the chain rule:         $\frac{d}{dx}[x^r] = \frac{d}{dx}[e^{r\ln(x)}]$

  • Outer function derivative is $e^{r\ln(x)}$.

  • Inner function derivative (where $r$ is a constant multiplier):         $\frac{d}{dx}[r\ln(x)] = r \times \frac{1}{x} = \frac{r}{x}$

  • Multiply outer derivative by inner derivative:         $\frac{d}{dx}[x^r] = e^{r\ln(x)} \times \frac{r}{x}$

  • Substitute $e^{r\ln(x)} = x^r$ back into the expression:         $\frac{d}{dx}[x^r] = x^r \times \frac{r}{x} = r \times \frac{x^r}{x} = r x^{r-1}$

• General Power Rule Statement:

  • For any real number exponent $r$:         $\frac{d}{dx}[x^r] = r x^{r-1}$

Advanced Multi-Rule Differentiation Problems

• Combining Chain Rule and Quotient Rule:

  • Problem Structure:         $y = \left(\frac{2x^2 - x^{-3}}{\sin(x) + \ln(x)}\right)^5$

  • Step 1: Apply Chain Rule to Outermost Power:         $\frac{dy}{dx} = 5 \left(\frac{2x^2 - x^{-3}}{\sin(x) + \ln(x)}\right)^4 \times \frac{d}{dx}\left[\frac{2x^2 - x^{-3}}{\sin(x) + \ln(x)}\right]$

  • Step 2: Apply Quotient Rule to Inner Fraction:

    • Numerator: $N(x) = 2x^2 - x^{-3}$

    • Numerator Derivative: $N'(x) = 4x + 3x^{-4}$

    • Denominator: $D(x) = \sin(x) + \ln(x)$

    • Denominator Derivative: $D'(x) = \cos(x) + \frac{1}{x}$

    • Quotient Rule Formula:             $\frac{d}{dx}\left[\frac{N(x)}{D(x)}\right] = \frac{N'(x)D(x) - N(x)D'(x)}{(D(x))^2}$

  • Step 3: Assemble Full Unsimplified Derivative:         $\frac{dy}{dx} = 5 \left(\frac{2x^2 - x^{-3}}{\sin(x) + \ln(x)}\right)^4 \times \frac{(4x + 3x^{-4})(\sin(x) + \ln(x)) - (2x^2 - x^{-3})\left(\cos(x) + \frac{1}{x}\right)}{(\sin(x) + \ln(x))^2}$

• Nested Chain Rule with Multiple Composition Layers:

  • Problem Structure:         $y = \sin^2(\ln(x^5 + x)) = [\sin(\ln(x^5 + x))]^2$

  • Layer Decomposition:

    1. Outer Power Layer: $f_1(u) = u^2$

    2. Trigonometric Layer: $f_2(v) = \sin(v)$

    3. Logarithmic Layer: $f_3(w) = \ln(w)$

    4. Polynomial Inner Layer: $f_4(x) = x^5 + x$

  • Step-by-Step Layer Differentiation:

    1. Derivative of Outer Power Layer: $2\sin(\ln(x^5 + x))$

    2. Derivative of Trigonometric Layer: $\cos(\ln(x^5 + x))$

    3. Derivative of Logarithmic Layer: $\frac{1}{x^5 + x}$

    4. Derivative of Polynomial Inner Layer: $5x^4 + 1$

  • Final Combined Product:         $\frac{dy}{dx} = 2\sin(\ln(x^5 + x)) \times \cos(\ln(x^5 + x)) \times \frac{1}{x^5 + x} \times (5x^4 + 1)$