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Video Title: Understanding Differentiation – A Simple Guide with Examples

Introduction (0:00 - 0:30)

• Visual: A simple animation of a graph with a curve.
• Voiceover:

  "Have you ever wondered how we find the rate of change of a function? This is where differentiation comes in. Differentiation helps us find the slope of a curve at any given point. In this video, we’ll break it down step by step, using simple examples."

What is Differentiation? (0:30 - 1:30)

• Visual: Show a car moving on a road with speed increasing.
• Voiceover:

  "Imagine you're driving a car, and you want to know how fast you're going at a specific moment. Differentiation allows us to calculate the instantaneous rate of change of a function, like velocity from position."

• Equation on screen:
  $f'(x) = \lim_{{h \to 0}} \frac{f(x+h) - f(x)}{h}$

  "This formula represents the derivative, which is the foundation of differentiation."

Basic Rules of Differentiation (1:30 - 3:30)

1. Power Rule

• Example:
  $\frac{d}{dx} x^n = n x^{n-1}$
  • Example: Differentiate $f(x) = x^3$
    • Answer: $f'(x) = 3x^2$
• Visual: Graph showing the slope changing as x increases.

2. Sum and Difference Rule

• Example:
  $\frac{d}{dx} [f(x) + g(x)] = f'(x) + g'(x)$
  • Example: $\frac{d}{dx} (x^2 + 3x) = 2x + 3$

3. Product Rule

• Example:
  $\frac{d}{dx} [u v] = u' v + u v'$
  • Example: $\frac{d}{dx} (x^2 \cdot \sin x) = 2x \sin x + x^2 \cos x$

4. Quotient Rule

• Example:
  $\frac{d}{dx} \left( \frac{u}{v} \right) = \frac{u' v - u v'}{v^2}$
  • Example: $\frac{d}{dx} \left( \frac{x^2}{x+1} \right)$

5. Chain Rule

• Example:
  $\frac{d}{dx} f(g(x)) = f'(g(x)) \cdot g'(x)$
  • Example: $\frac{d}{dx} (\sin(x^2)) = \cos(x^2) \cdot 2x$

Applications of Differentiation (3:30 - 5:00)

1. Finding Slopes of Curves – Show how a tangent line touches a curve.
2. Optimization (Maxima and Minima) – Show a hill and valley.
3. Physics (Velocity & Acceleration) – Differentiate position to get velocity, then acceleration.

Conclusion (5:00 - 5:30)

• Visual: A summary table of rules and a graph with a tangent line.
• Voiceover:

  "Differentiation is a powerful tool in mathematics and science. By learning these basic rules, you can solve complex problems in calculus, physics, and engineering. If you want to practice more, try solving these problems on your own!"

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Related Questions:

1. How do you differentiate logarithmic and exponential functions?
2. What are higher-order derivatives, and how are they useful?
3. How do you find critical points using derivatives?
4. How is differentiation applied in economics?
5. What is implicit differentiation, and when is it used?

Tip: Always check if a function is differentiable before applying the rules!