Derivatives: Rules, Graphs, and Business Applications

TAKEAWAYS:

• Derivatives measure how fast a function’s output changes with respect to its input (slope of a tangent line).
• First principles (limit definition) and shortcut rules (power, product, quotient, chain) help compute derivatives quickly.
• Graphical interpretation: A derivative tells you if a function is increasing/decreasing, concave up/down, or has maxima/minima.
• Business uses: Derivatives model profit maximization, cost minimization, and revenue optimization.
• Exam focus: Know how to apply rules to real-world functions (e.g., cost, revenue, profit) and interpret results.

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1. What is a Derivative?

A derivative tells us how fast a quantity changes at any point. For example:

• If $y=f(x)$ represents a company’s profit over time, $f^{\prime }(x)$ tells you how fast profit is increasing or decreasing right now.

Key Idea: The Limit Definition (First Principles)

The derivative of $f(x)$ at $x=a$ is: $$f^{\prime }(a)=\underset{h\to 0}{\lim }\frac{f(a+h)-f(a)}{h}$$ This is the slope of the tangent line at $x=a$.

Example 1: Find $f^{\prime }(x)$ for $f(x)=x^2$ using first principles. Solution: $$f^{\prime }(x)=\underset{h\to 0}{\lim }\frac{(x+h{)}^2-x^2}{h}=\underset{h\to 0}{\lim }\frac{2xh+h^2}{h}=\underset{h\to 0}{\lim }(2x+h)=2x$$ Answer: $f^{\prime }(x)=2x$.

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2. Basic Rules of Differentiation

Instead of using first principles every time, we use shortcut rules:

Example 2: Differentiate $f(x)=4x^3-2x+7$. Solution: $$f^{\prime }(x)=4\cdot 3x^2-2\cdot 1+0=12x^2-2$$ Answer: $f^{\prime }(x)=12x^2-2$.

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3. Product, Quotient, and Chain Rules

For more complex functions, we use:

Product Rule

If $f(x)=u(x)\cdot v(x)$, then: $$f^{\prime }(x)=u^{\prime }(x)v(x)+u(x)v^{\prime }(x)$$

Example 3: Differentiate $f(x)=x^2\cdot \sin x$. Solution: Let $u=x^2$, $v=\sin x$. $$u^{\prime }=2x,v^{\prime }=\cos x$$ $$f^{\prime }(x)=(2x)(\sin x)+(x^2)(\cos x)=2x\sin x+x^2\cos x$$ Answer: $f^{\prime }(x)=2x\sin x+x^2\cos x$.

Quotient Rule

If $f(x)=\frac{u(x)}{v(x)}$, then: $$f^{\prime }(x)=\frac{u^{\prime }(x)v(x)-u(x)v^{\prime }(x)}{[v(x){]}^2}$$

Example 4: Differentiate $f(x)=\frac{x}{x^2+1}$. Solution: Let $u=x$, $v=x^2+1$. $$u^{\prime }=1,v^{\prime }=2x$$ $$f^{\prime }(x)=\frac{(1)(x^2+1)-(x)(2x)}{(x^2+1{)}^2}=\frac{1-x^2}{(x^2+1{)}^2}$$ Answer: $f^{\prime }(x)=\frac{1-x^2}{(x^2+1{)}^2}$.

Chain Rule

If $f(x)=g(h(x))$, then: $$f^{\prime }(x)=g^{\prime }(h(x))\cdot h^{\prime }(x)$$

Example 5: Differentiate $f(x)=\sin (3x^2)$. Solution: Let $g(u)=\sin u$, $h(x)=3x^2$. $$g^{\prime }(u)=\cos u,h^{\prime }(x)=6x$$ $$f^{\prime }(x)=\cos (3x^2)\cdot 6x=6x\cos (3x^2)$$ Answer: $f^{\prime }(x)=6x\cos (3x^2)$.

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4. Graphical Interpretation of Derivatives

The derivative $f^{\prime }(x)$ tells us:

• Sign of $f^{\prime }(x)$:
  • $f^{\prime }(x)>0$: Function is increasing.
  • $f^{\prime }(x)<0$: Function is decreasing.
• Critical Points: Where $f^{\prime }(x)=0$ or $f^{\prime }(x)$ is undefined (possible maxima/minima).
• Concavity:
  • $f^{\prime \prime }(x)>0$: Concave up (like a cup $\cup$).
  • $f^{\prime \prime }(x)<0$: Concave down (like a cap $\cap$).

Example 6: Analyze $f(x)=x^3-3x^2$.

1. Find $f^{\prime }(x)$: $$f^{\prime }(x)=3x^2-6x$$
2. Find critical points: $$3x^2-6x=0⟹x(3x-6)=0⟹x=0\text{ or }x=2$$
3. Determine increasing/decreasing:
   • For $x<0$: $f^{\prime }(x)>0$ (increasing).
   • For $0<x<2$: $f^{\prime }(x)<0$ (decreasing).
   • For $x>2$: $f^{\prime }(x)>0$ (increasing).
4. Concavity (find $f^{\prime \prime }(x)$): $$f^{\prime \prime }(x)=6x-6$$
   • $f^{\prime \prime }(x)>0$ when $x>1$ (concave up).
   • $f^{\prime \prime }(x)<0$ when $x<1$ (concave down).

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5. Applications in Business

Derivatives help solve real-world problems like:

1. Profit Maximization: Find where ${\text{Profit}}^{\prime }(x)=0$.
2. Cost Minimization: Find where ${\text{Cost}}^{\prime }(x)=0$.
3. Revenue Optimization: Find where ${\text{Revenue}}^{\prime }(x)=0$.

Example 7: A company’s profit function is $P(x)=-2x^2+100x-1000$. Find the production level $x$ that maximizes profit. Solution:

1. Find $P^{\prime }(x)$: $$P^{\prime }(x)=-4x+100$$
2. Set $P^{\prime }(x)=0$: $$-4x+100=0⟹x=25$$
3. Verify it’s a maximum (second derivative test): $$P^{\prime \prime }(x)=-4<0⟹\text{Maximum at }x=25$$ Answer: Produce 25 units for maximum profit.

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6. Higher-Order Derivatives

The second derivative $f^{\prime \prime }(x)$ tells us about concavity and inflection points.

• If $f^{\prime \prime }(x)>0$: Concave up.
• If $f^{\prime \prime }(x)<0$: Concave down.
• If $f^{\prime \prime }(x)=0$: Possible inflection point.

Example 8: Find $f^{\prime \prime }(x)$ for $f(x)=x^4-2x^3+x$. Solution:

1. First derivative: $$f^{\prime }(x)=4x^3-6x^2+1$$
2. Second derivative: $$f^{\prime \prime }(x)=12x^2-12x$$ Answer: $f^{\prime \prime }(x)=12x^2-12x$.

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Exam Tip

1. Memorize the rules: Power, product, quotient, and chain rules are must-know.
2. Practice first principles: NEB often tests this for simple functions.
3. Graph interpretation: Always sketch the function and its derivative to understand behavior.
4. Business applications: Know how to find maxima/minima for profit, cost, and revenue functions.
5. Second derivative test: Use it to confirm maxima/minima in optimization problems.

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NEB-Style Questions

Short Answer:

1. Find the derivative of $f(x)=3x^4-5x^2+7$ using the power rule.
2. Differentiate $f(x)=\frac{x^2+1}{x}$ using the quotient rule.
3. If $f(x)=e^{2x}$, find $f^{\prime }(x)$ using the chain rule.

Long Answer: 4. A company’s cost function is $C(x)=0.5x^2+100x+500$. Find the production level $x$ that minimizes cost. 5. Given $f(x)=x^3-6x^2+9x$, find:

• All critical points.
• Intervals where $f(x)$ is increasing/decreasing.
• Concavity and inflection points.

Graphical Question: 6. Sketch the graph of $f(x)=x^2-4x+3$ and its derivative $f^{\prime }(x)$. Label:

• Where $f(x)$ is increasing/decreasing.
• The critical point and its nature (max/min).

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Note: Always show every step in your exam answers. Partial credit is given for correct intermediate steps!