Interactive Worksheet

Differentiation Notation, Rules & Applications

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1 Foundations

Foundations & Notation

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Calculus focuses on two main ideas: differentiation and integration. Differentiation tells us how a function changes at each point. Integration does the opposite—it tells us how those changes add up or accumulate.

Derivative Notation

Finding a derivative is called .

If $y = f(x)$ is a function whose derivative exists, the first derivative can be written in different notations:

Prime Notation

$y' =$

Differential Notation

$y' =$ $= \frac{d}{dx}f(x)$

For a function $f$, the first derivative of $f$ at $x = a$, denoted by $f'(a)$, represents the rate of change of $f$ at that specific value.

Graphically, $f'(a)$ is the of the to the graph of $f$ at the point $(a, f(a))$.

Interactive Tool: Tangent Explorer

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2 Basic Rules

Building Blocks & Basic Rules

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Memorizing these basic derivatives is crucial for everything else in calculus. Fill in the resulting derivative $f'(x)$ for each base function.

Rule Name Function $f(x)$ Derivative $f'(x)$
Constant $c$
Power $x^n$
Identity $x$
Exponential $e^x$
Natural Log $\ln(x)$

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3 Complex Rules

Complex & Combined Rules

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The Product Rule

If $F(x) = f(x)g(x)$, then $F'(x) = f'(x)g(x) +$

Example: For $k(x) = 7e^x(x^3-2)$, the derivative is:
$k'(x) = 7e^x(x^3-2) + $ $\cdot (7e^x)$

The Quotient Rule

If $F(x) = \frac{f(x)}{g(x)}$, then $F'(x) = \frac{f'(x)g(x) - g'(x)f(x)}{[g(x)]^2}$

Example: $h(s) = \frac{-6s^{1/2}}{3s^2+2s}$. Identify the derivatives of top and bottom:

  • Numerator deriv. $f'(s) = $
  • Denominator deriv. $g'(s) = $

The Chain Rule

If $F(x) = f(g(x))$, then $F'(x) = f'(g(x)) \cdot$

"Derivative of the outside, leave the inside alone, times derivative of the inside."

1. $\frac{d}{dx} \ln(10-7x)$

$= \frac{1}{10-7x} \cdot ($ $)$

2. $\frac{d}{dx} e^{0.01x^3}$

$= e^{0.01x^3} \cdot ($ $)$

Knowledge Check

Apply Product, Quotient, or Chain rules.

4 Real-World Applications

Real-World Applications

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Calculus isn't just about formulas; it models the real world. A derivative represents the rate of change of any function, whether that's velocity in physics, marginal cost in business, or population growth in biology.

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