Maths1_Week8_Derivatives

Maths 1 Week 8: Derivatives & Continuity

0. Prerequisites

NOTE

What you need to know:

• Slope: Rise over Run ($\frac{\Delta y}{\Delta x}$).
• Limits: ${\lim }_{h\to 0}$.
• Tangent: A line that just touches a curve.

Quick Refresher

• Derivative ($f^{\prime }(x)$): The slope of the tangent line at any point $x$.
• Rate of Change: How fast $y$ is changing with respect to $x$.

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1. Core Concepts

1.1 Rules of Differentiation

1. Power Rule: $\frac{d}{dx}(x^n)=n{x}^{n-1}$.
2. Constant Multiple: $\frac{d}{dx}(cf(x))=c{f}^{\prime }(x)$.
3. Sum Rule: $(f+g{)}^{\prime }=f^{\prime }+g^{\prime }$.
4. Product Rule: $(fg{)}^{\prime }=f^{\prime }g+f{g}^{\prime }$.
5. Quotient Rule: $(\frac{f}{g}{)}^{\prime }=\frac{f^{\prime }g-f{g}^{\prime }}{g^2}$.
6. Chain Rule: $\frac{d}{dx}f(g(x))=f^{\prime }(g(x))\cdot g^{\prime }(x)$.

1.2 Tangent Line & Approximation

• Equation of Tangent: At point $(a,f(a))$: $y-f(a)=f^{\prime }(a)(x-a)$
• Linear Approximation: Near $a$, the function looks like the tangent line. $L(x)=f(a)+f^{\prime }(a)(x-a)$

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2. Pattern Analysis & Goated Solutions

Pattern 1: The Chain Rule (Composite Functions)

Context: “Find derivative of $y=(3x^2+1{)}^5$.”

TIP

Mental Algorithm:

1. Identify Layers: Outer function (Power 5) and Inner function ($3x^2+1$).
2. Differentiate Outer: Bring power down, keep inside SAME. $5(\dots {)}^4$.
3. Multiply by Inner Derivative: Multiply by derivative of inside ($6x$).

Example (Detailed Solution)

Problem: Find $f^{\prime }(x)$ for $f(x)=\sqrt{x^2+5}$. Solution:

1. Rewrite: $f(x)=(x^2+5{)}^{1/2}$.
2. Outer Derivative:
   • Power rule on $(\dots {)}^{1/2}$.
   • $\frac{1}{2}(\dots {)}^{-1/2}$.
   • $\frac{1}{2}(x^2+5{)}^{-1/2}$.
3. Inner Derivative:
   • Inside is $x^2+5$.
   • Derivative is $2x$.
4. Combine:
   • $f^{\prime }(x)=\frac{1}{2}(x^2+5{)}^{-1/2}\cdot (2x)$.
   • Simplify: $\frac{2x}{2\sqrt{x^2+5}}=\frac{x}{\sqrt{x^2+5}}$. Answer: $\frac{x}{\sqrt{x^2+5}}$.

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Pattern 2: Equation of Tangent Line

Context: “Find the equation of the tangent to $y=x^2$ at $x=3$.”

TIP

Mental Algorithm:

1. Point: Find $y$ when $x=a$. Point $(a,f(a))$.
2. Slope: Find $f^{\prime }(x)$ and plug in $x=a$. Slope $m=f^{\prime }(a)$.
3. Line: Use Point-Slope form: $y-y_1=m(x-x_1)$.

Example (Detailed Solution)

Problem: Find tangent to $f(x)=x^3-x$ at $x=2$. Solution:

1. Find Point:
   • $f(2)=2^3-2=8-2=6$.
   • Point is $(2,6)$.
2. Find Slope:
   • $f^{\prime }(x)=3x^2-1$.
   • $f^{\prime }(2)=3(2{)}^2-1=3(4)-1=11$.
   • Slope $m=11$.
3. Equation:
   • $y-6=11(x-2)$.
   • $y-6=11x-22$.
   • $y=11x-16$. Answer: $y=11x-16$.

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Pattern 3: Optimization (Max/Min)

Context: “Find dimensions to maximize area.”

TIP

Mental Algorithm:

1. Function: Write the formula for what you want to maximize (Area $A(x)$).
2. Derivative: Find $A^{\prime }(x)$.
3. Critical Point: Set $A^{\prime }(x)=0$ and solve for $x$.
4. Verify: Check if it’s a max (using logic or 2nd derivative).

Example (Detailed Solution)

Problem: Maximize product of two numbers that sum to 20. Solution:

1. Variables: Let numbers be $x$ and $y$. $x+y=20⟹y=20-x$.
2. Function: Product $P=x\cdot y=x(20-x)=20x-x^2$.
3. Derivative:
   • $P^{\prime }(x)=20-2x$.
4. Critical Point:
   • Set $P^{\prime }(x)=0$.
   • $20-2x=0⟹2x=20⟹x=10$.
5. Result:
   • If $x=10$, then $y=10$.
   • Product = 100. Answer: The numbers are 10 and 10.

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3. Practice Exercises

1. Derivative: Find $y^{\prime }$ for $y=x^5+3x^2$.
   • Hint: $5x^4+6x$.
2. Tangent: Slope of tangent to $y=\sqrt{x}$ at $x=4$.
   • Hint: $y^{\prime }=\frac{1}{2\sqrt{x}}$. At 4, slope is $1/4$.
3. Chain Rule: Derivative of $(2x+1{)}^3$.
   • Hint: $3(2x+1{)}^2\cdot 2=6(2x+1{)}^2$.

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🧠 Level Up: Advanced Practice

Question 1: The Chain Rule Trap

Problem: Find derivative of $y={\sin }^3(2x^2+1)$. Logic:

1. Outer Layer: Power function $u^3\to 3u^2$.
   • $3{\sin }^2(2x^2+1)$.
2. Middle Layer: Sine function $\sin (v)\to \cos (v)$.
   • $\cos (2x^2+1)$.
3. Inner Layer: Polynomial $2x^2+1\to 4x$.
4. Combine: $3{\sin }^2(2x^2+1)\cdot \cos (2x^2+1)\cdot 4x$. Answer: $12x{\sin }^2(2x^2+1)\cos (2x^2+1)$.

Question 2: Optimization (The Box Problem)

Problem: Square sheet of side 12cm. Cut squares of side $x$ from corners and fold up. Max Volume? Logic:

1. Dimensions: Length $12-2x$, Width $12-2x$, Height $x$.
2. Volume: $V(x)=x(12-2x{)}^2=x(144-48x+4x^2)=4x^3-48x^2+144x$.
3. Derivative: $V^{\prime }(x)=12x^2-96x+144$.
4. Critical Points: $12(x^2-8x+12)=0⟹12(x-6)(x-2)=0$.
   • $x=6$ or $x=2$.
5. Check:
   • If $x=6$, Width $=12-12=0$. Volume 0. (Min).
   • If $x=2$, Width $=12-4=8$. Volume $2\times 8\times 8=128$. (Max). Answer: Max Volume is 128 cm${}^3$ at $x=2$.

Question 3: Tangent Parallel to X-Axis

Problem: Find points on $y=x^3-3x$ where tangent is horizontal. Logic:

1. Slope: Horizontal means $m=0$.
2. Derivative: $y^{\prime }=3x^2-3$.
3. Set to 0: $3x^2-3=0⟹x^2=1⟹x=\pm 1$.
4. Points:
   • $x=1⟹y=1-3=-2$. Point $(1,-2)$.
   • $x=-1⟹y=-1+3=2$. Point $(-1,2)$. Answer: $(1,-2)$ and $(-1,2)$.