Introduction to Derivatives

What is a Derivative?

Derivative: Instantaneous rate of change of a function

Measures: How fast function is changing at a specific point

Geometric meaning: Slope of tangent line to curve

Notation:

• f'(x) (read "f prime of x")
• dy/dx
• df/dx

Average Rate of Change

Review: Average rate of change over interval [a, b]

Formula: [f(b) - f(a)] / (b - a)

Geometric meaning: Slope of secant line through (a, f(a)) and (b, f(b))

Example: Average Rate

f(x) = x²

Average rate from x = 1 to x = 3:

[f(3) - f(1)] / (3 - 1) = [9 - 1] / 2 = 8/2 = 4

Average rate of change: 4

Instantaneous Rate of Change

Instantaneous rate: Rate at exact point (not over interval)

Find by: Making interval smaller and smaller (limit)

Definition of derivative:

f'(x) = lim(h→0) [f(x+h) - f(x)] / h

Where h is small change in x

Example: Find Derivative Using Limit

f(x) = x²

Find f'(x):

f'(x) = lim(h→0) [(x+h)² - x²] / h
      = lim(h→0) [x² + 2xh + h² - x²] / h
      = lim(h→0) [2xh + h²] / h
      = lim(h→0) [h(2x + h)] / h
      = lim(h→0) (2x + h)
      = 2x

Derivative: f'(x) = 2x

At x = 3: f'(3) = 6 (instantaneous rate)

Derivative as Slope of Tangent Line

Tangent line: Line that just touches curve at one point

Slope of tangent at x = a: f'(a)

Equation of tangent line at (a, f(a)):

y - f(a) = f'(a)(x - a)

Example: Find Tangent Line

f(x) = x², find tangent at x = 2

Point: (2, 4)

Slope: f'(2) = 2(2) = 4

Tangent line:

y - 4 = 4(x - 2)
y - 4 = 4x - 8
y = 4x - 4

Power Rule

For f(x) = x^n:

f'(x) = n·x^(n-1)

Most important derivative rule!

Example 1: Power Rule

f(x) = x³

f'(x) = 3x²

Example 2: Different Powers

f(x) = x⁵ f'(x) = 5x⁴

g(x) = x⁴ g'(x) = 4x³

h(x) = x h'(x) = 1·x⁰ = 1

Example 3: Constant

f(x) = 7 (constant function)

f'(x) = 0 (derivative of constant is 0)

Constant Multiple Rule

For f(x) = c·g(x):

f'(x) = c·g'(x)

Constant comes out front

Example: Constant Multiple

f(x) = 5x³

f'(x) = 5·(3x²) = 15x²

Sum and Difference Rules

For f(x) = g(x) + h(x):

f'(x) = g'(x) + h'(x)

For f(x) = g(x) - h(x):

f'(x) = g'(x) - h'(x)

Take derivative of each term

Example: Polynomial

f(x) = 3x⁴ - 2x³ + 5x - 7

f'(x) = 3(4x³) - 2(3x²) + 5(1) - 0 f'(x) = 12x³ - 6x² + 5

Special Derivatives

f(x) = sin(x) → f'(x) = cos(x)

f(x) = cos(x) → f'(x) = -sin(x)

f(x) = e^x → f'(x) = e^x (exponential function)

f(x) = ln(x) → f'(x) = 1/x (natural log)

Example: Trig Derivative

f(x) = 2sin(x) + 3cos(x)

f'(x) = 2cos(x) + 3(-sin(x)) f'(x) = 2cos(x) - 3sin(x)

Product Rule

For f(x) = g(x)·h(x):

f'(x) = g'(x)·h(x) + g(x)·h'(x)

"First derivative times second, plus first times second derivative"

Example: Product Rule

f(x) = x²·sin(x)

f'(x) = (2x)·sin(x) + x²·cos(x) f'(x) = 2x·sin(x) + x²·cos(x)

Quotient Rule

For f(x) = g(x)/h(x):

f'(x) = [g'(x)·h(x) - g(x)·h'(x)] / [h(x)]²

"Low dee-high minus high dee-low, over low squared"

Example: Quotient Rule

f(x) = x²/(x+1)

f'(x) = [(2x)(x+1) - (x²)(1)] / (x+1)² f'(x) = [2x² + 2x - x²] / (x+1)² f'(x) = [x² + 2x] / (x+1)²

Chain Rule

For composite f(g(x)):

[f(g(x))]' = f'(g(x))·g'(x)

"Derivative of outside times derivative of inside"

Example: Chain Rule

f(x) = (x² + 1)³

Outside: u³, inside: u = x² + 1

f'(x) = 3(x² + 1)²·(2x) f'(x) = 6x(x² + 1)²

Interpreting the Derivative

f'(x) > 0: Function increasing

f'(x) < 0: Function decreasing

f'(x) = 0: Horizontal tangent (possible max/min)

Example: Where Increasing?

f(x) = x² - 4x + 3

f'(x) = 2x - 4

Increasing when f'(x) > 0:

2x - 4 > 0
x > 2

Function increasing for x > 2

Real-World Applications

Physics: Velocity is derivative of position

• If s(t) = position, then v(t) = s'(t) = velocity
• Acceleration a(t) = v'(t) = s''(t)

Economics: Marginal cost, marginal revenue

Biology: Population growth rates

Engineering: Rates of change in systems

Example: Velocity

Position: s(t) = 16t² feet (falling object)

Velocity:

v(t) = s'(t) = 32t ft/s

At t = 2 seconds:

v(2) = 32(2) = 64 ft/s

Object falling at 64 ft/s after 2 seconds

Higher-Order Derivatives

Second derivative: f''(x) = (f'(x))'

Measures: Rate of change of rate of change

Concavity:

• f''(x) > 0: Concave up (curve opens upward)
• f''(x) < 0: Concave down (curve opens downward)

Example: Second Derivative

f(x) = x³

f'(x) = 3x²

f''(x) = 6x

Critical Points

Critical point: Where f'(x) = 0 or undefined

Possible locations of: Local max, local min, or inflection point

Example: Find Critical Points

f(x) = x³ - 3x² + 2

f'(x) = 3x² - 6x

Set equal to 0:

3x² - 6x = 0
3x(x - 2) = 0
x = 0 or x = 2

Critical points at x = 0 and x = 2

Practice