Differentiation: Rules, Techniques & Applications (Single Variable)

Core Concepts & Definitions

1.1 Definition of the Derivative

The derivative of a function $f(x)$ at a point $x=a$, denoted $f^{\prime }(a)$ or $\frac{df}{dx}{|}_{x=a}$, is defined as the limit of the difference quotient as $h\to 0$:

$$f^{\prime }(a)=\underset{h\to 0}{\lim }\frac{f(a+h)-f(a)}{h}$$

Key Observations:

• Geometric Interpretation: The derivative represents the slope of the tangent line to the curve $y=f(x)$ at $x=a$.
• Physical Interpretation: For motion functions $s(t)$, $s^{\prime }(t)$ gives instantaneous velocity.
• Existence: A function must be continuous at $x=a$ for the derivative to exist (though continuity does not guarantee differentiability).

Worked Example: Find $f^{\prime }(x)$ for $f(x)=x^2$ using the limit definition. Solution: $$f^{\prime }(x)=\underset{h\to 0}{\lim }\frac{(x+h{)}^2-x^2}{h}=\underset{h\to 0}{\lim }\frac{2xh+h^2}{h}=2x$$

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1.2 Differentiability vs. Continuity

Why It Matters:

• Non-differentiable points occur at:
  • Corners (e.g., $f(x)=|x|$ at $x=0$).
  • Cusps (e.g., $f(x)=x^{2/3}$ at $x=0$).
  • Vertical tangents (e.g., $f(x)=\sqrt[3]{x}$ at $x=0$).

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Fundamental Rules of Differentiation

2.1 Basic Rules

Worked Example: Differentiate $f(x)=4x^3-2x+9$. Solution: $$f^{\prime }(x)=12x^2-2$$

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2.2 Product, Quotient, and Chain Rules

Product Rule

For $f(x)=u(x)\cdot v(x)$: $$f^{\prime }(x)=u^{\prime }(x)v(x)+u(x)v^{\prime }(x)$$ Example: Differentiate $f(x)=x^2\sin x$. Solution: $$f^{\prime }(x)=2x\sin x+x^2\cos x$$

Quotient Rule

For $f(x)=\frac{u(x)}{v(x)}$: $$f^{\prime }(x)=\frac{u^{\prime }(x)v(x)-u(x)v^{\prime }(x)}{[v(x){]}^2}$$ Example: Differentiate $f(x)=\frac{x}{\sin x}$. Solution: $$f^{\prime }(x)=\frac{\sin x-x\cos x}{{\sin }^{2}x}$$

Chain Rule (Composite Functions)

For $f(x)=g(h(x))$: $$f^{\prime }(x)=g^{\prime }(h(x))\cdot h^{\prime }(x)$$ Example: Differentiate $f(x)=\sin (3x^2)$. Solution: $$f^{\prime }(x)=\cos (3x^2)\cdot 6x=6x\cos (3x^2)$$

Comparison Table:

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Implicit and Logarithmic Differentiation

3.1 Implicit Differentiation

Used when $y$ is not isolated (e.g., $x^2+y^2=25$). Steps:

1. Differentiate both sides w.r.t. $x$, treating $y$ as $y(x)$.
2. Solve for $\frac{dy}{dx}$.

Worked Example: Find $\frac{dy}{dx}$ for $x^2+y^2=25$. Solution: $$2x+2y\frac{dy}{dx}=0⟹\frac{dy}{dx}=-\frac{x}{y}$$

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3.2 Logarithmic Differentiation

Useful for functions like $f(x)=x^x$ or $f(x)=\sqrt{\frac{x+1}{x-1}}$. Steps:

1. Take the natural log: $\ln f(x)$.
2. Differentiate implicitly.
3. Solve for $f^{\prime }(x)$.

Worked Example: Differentiate $f(x)=x^x$. Solution: $$\ln f(x)=x\ln x⟹\frac{f^{\prime }(x)}{f(x)}=\ln x+1⟹f^{\prime }(x)=x^x(\ln x+1)$$

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Higher-Order Derivatives

The second derivative $f^{\prime \prime }(x)$ is the derivative of $f^{\prime }(x)$. Applications:

• Concavity: $f^{\prime \prime }(x)>0$ → concave up; $f^{\prime \prime }(x)<0$ → concave down.
• Inflection Points: Where $f^{\prime \prime }(x)=0$ or is undefined.

Worked Example: Find $f^{\prime \prime }(x)$ for $f(x)=x^3-2x^2+x$. Solution: $$\begin{gathered} f^{\prime }(x)=3x^2-4x+1 \\ {f}^{\prime \prime }(x)=6x-4 \end{gathered}$$

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Curve Sketching Using Derivatives

To sketch $y=f(x)$, analyze:

1. Domain: Where $f(x)$ is defined.
2. Intercepts: $f(0)$ (y-intercept), $f(x)=0$ (x-intercepts).
3. Symmetry: Even/odd functions.
4. First Derivative Test:
   • $f^{\prime }(x)>0$ → increasing.
   • $f^{\prime }(x)<0$ → decreasing.
   • Critical points: $f^{\prime }(x)=0$ or undefined.
5. Second Derivative Test:
   • Concavity and inflection points.
6. Asymptotes:
   • Vertical: Where $f(x)\to \pm \infty$.
   • Horizontal/Oblique: Limits as $x\to \pm \infty$.

Worked Example: Sketch $f(x)=x^3-3x^2$. Solution:

1. Derivatives: $$\begin{gathered} f^{\prime }(x)=3x^2-6x=3x(x-2) \\ {f}^{\prime \prime }(x)=6x-6 \end{gathered}$$
2. Critical Points: $x=0,2$.
3. Intervals:
   • $f^{\prime }(x)>0$ for $x<0$ or $x>2$ → increasing.
   • $f^{\prime }(x)<0$ for $0<x<2$ → decreasing.
4. Concavity:
   • $f^{\prime \prime }(x)>0$ for $x>1$ → concave up.
   • $f^{\prime \prime }(x)<0$ for $x<1$ → concave down.
5. Inflection Point: $x=1$.
6. Sketch:
   • Roots at $x=0,3$.
   • Local max at $x=0$, local min at $x=2$.

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Exam Tip

Common Pitfalls & Strategies

1. Limit Definition:

   • Always simplify the difference quotient before taking the limit.
   • Example Mistake: Forgetting to factor $h$ in $\frac{(x+h{)}^2-x^2}{h}$.

2. Chain Rule:

   • TU Exam Trick: For nested functions, work from the outside in.
   • Example: $\frac{d}{dx}\sin (e^{x^2})=\cos (e^{x^2})\cdot e^{x^2}\cdot 2x$.

3. Implicit Differentiation:

   • Key Step: Remember $\frac{d}{dx}[y^n]=n{y}^{n-1}\frac{dy}{dx}$.
   • Example: For $x^{2}y+y^3=5$, differentiate $y^3$ as $3y^2\frac{dy}{dx}$.

4. Higher-Order Derivatives:

   • Shortcut: For polynomials, reduce the exponent by 1 each time.
   • Example: $f(x)=x^4⟹f^{\prime \prime }(x)=12x^2$.

5. Curve Sketching:

   • Must Include: All critical points, intercepts, and concavity changes.
   • TU Expectation: Label increasing/decreasing intervals and inflection points.

6. Logarithmic Differentiation:

   • When to Use: For functions like $x^x$, $\sqrt{x}$, or products/quotients with variables in exponents.
   • Avoid: Applying it to simple polynomials (overcomplicates).

High-Scoring Techniques

• Show All Steps: Partial credit is given for correct intermediate steps.
• Verify Results: Plug in a test point (e.g., $x=1$) to check your derivative.
• Graphical Intuition: Sketch a rough graph before calculating derivatives to guide your work.
• Unit Consistency: In applied problems, ensure units match (e.g., $\frac{dy}{dx}$ should have consistent units).

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Practice Problems (TU-Style)

1. Find $\frac{dy}{dx}$ for $x^{2}y+\ln y=x$.
2. Differentiate $f(x)={\left(\frac{x+1}{x-1}\right)}^{1/2}$ using logarithmic differentiation.
3. For $f(x)=x{e}^x$, find $f^{\prime \prime }(x)$ and determine concavity.
4. Sketch the curve $y=x^3-6x^2+9x$, labeling all critical points and inflection points.