Grade: Grade 12 Subject: Mathematics Unit: Calculus Introduction SAT: AdvancedMath ACT: Math

Introduction to Derivatives

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What is a Derivative?

The derivative is one of the most powerful concepts in mathematics. It measures the instantaneous rate of change of a function - essentially, how fast something is changing at any given moment. This concept forms the foundation of calculus and has countless applications in physics, engineering, economics, and beyond.

The Derivative

The derivative of a function f(x) at a point x = a is defined as:

f'(a) = lim (h→0) [f(a + h) - f(a)] / h

This represents the slope of the tangent line to the curve at that point, or equivalently, the instantaneous rate of change.

Notation for Derivatives

NotationRead AsWhen to Use
f'(x)"f prime of x"General function notation
dy/dx"dee y dee x"Leibniz notation, emphasizes the ratio
d/dx [f(x)]"derivative of f with respect to x"Operator notation
y'"y prime"Quick notation when context is clear

Basic Derivative Rules

Constant Rule

d/dx [c] = 0

The derivative of any constant is zero (constants don't change).

Power Rule

d/dx [x^n] = n·x^(n-1)

Multiply by the exponent, then reduce the exponent by 1.

Example: d/dx [x^5] = 5x^4

Constant Multiple Rule

d/dx [c·f(x)] = c·f'(x)

Constants can be "pulled out" of derivatives.

Example: d/dx [3x^4] = 3·4x^3 = 12x^3

Sum and Difference Rules

d/dx [f(x) ± g(x)] = f'(x) ± g'(x)

The derivative of a sum/difference is the sum/difference of derivatives.

Derivatives of Common Functions

Function f(x)Derivative f'(x)
x^nn·x^(n-1)
e^xe^x
ln(x)1/x
sin(x)cos(x)
cos(x)-sin(x)
tan(x)sec²(x)
a^xa^x · ln(a)

The Product Rule

Product Rule

For two functions f(x) and g(x):

d/dx [f(x)·g(x)] = f'(x)·g(x) + f(x)·g'(x)

Remember: "first times derivative of second, plus second times derivative of first"

The Quotient Rule

Quotient Rule

For a quotient f(x)/g(x):

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

Remember: "low d-high minus high d-low, all over low squared"

The Chain Rule

Chain Rule

For composite functions f(g(x)):

d/dx [f(g(x))] = f'(g(x)) · g'(x)

Take the derivative of the outer function (keeping inner function as is), then multiply by the derivative of the inner function.

Interpreting the Derivative

What Does f'(x) Tell Us?

  • f'(x) > 0: f is increasing at that point
  • f'(x) < 0: f is decreasing at that point
  • f'(x) = 0: f has a horizontal tangent (possible max/min)
  • |f'(x)| large: f is changing rapidly
  • |f'(x)| small: f is changing slowly

Tangent Lines

Equation of Tangent Line

At point (a, f(a)), the tangent line has slope f'(a):

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

Or: y = f(a) + f'(a)(x - a)

Examples

Example 1: Using Basic Rules

Problem: Find the derivative of f(x) = 4x^3 - 2x^2 + 5x - 7

Solution:

Apply rules term by term:

f'(x) = d/dx[4x^3] - d/dx[2x^2] + d/dx[5x] - d/dx[7]

f'(x) = 4(3x^2) - 2(2x^1) + 5(1x^0) - 0

f'(x) = 12x^2 - 4x + 5

Example 2: Product Rule

Problem: Find the derivative of f(x) = (2x + 1)(x^2 - 3)

Solution:

Let u = 2x + 1 and v = x^2 - 3

u' = 2 and v' = 2x

Product Rule: f'(x) = u'v + uv'

f'(x) = (2)(x^2 - 3) + (2x + 1)(2x)

f'(x) = 2x^2 - 6 + 4x^2 + 2x

f'(x) = 6x^2 + 2x - 6

Example 3: Quotient Rule

Problem: Find the derivative of f(x) = (3x + 2)/(x - 1)

Solution:

Let u = 3x + 2 (high) and v = x - 1 (low)

u' = 3 and v' = 1

Quotient Rule: f'(x) = (v·u' - u·v')/v^2

f'(x) = [(x - 1)(3) - (3x + 2)(1)] / (x - 1)^2

f'(x) = [3x - 3 - 3x - 2] / (x - 1)^2

f'(x) = -5/(x - 1)^2

Example 4: Chain Rule

Problem: Find the derivative of f(x) = (3x + 1)^5

Solution:

Outer function: u^5, Inner function: u = 3x + 1

Derivative of outer: 5u^4

Derivative of inner: 3

Chain Rule: f'(x) = 5(3x + 1)^4 · 3

f'(x) = 15(3x + 1)^4

Example 5: Finding Tangent Line

Problem: Find the equation of the tangent line to f(x) = x^2 - 2x + 3 at x = 2.

Solution:

Step 1: Find the point of tangency

f(2) = (2)^2 - 2(2) + 3 = 4 - 4 + 3 = 3

Point: (2, 3)

Step 2: Find the slope (derivative at x = 2)

f'(x) = 2x - 2

f'(2) = 2(2) - 2 = 2

Step 3: Write the tangent line equation

y - 3 = 2(x - 2)

y = 2x - 4 + 3

y = 2x - 1

Practice

Apply derivative rules to find the derivatives of these functions.

1. Find f'(x) if f(x) = 5x^4 - 3x^2 + 7

A) 20x^3 - 6x   B) 20x^3 - 6x + 7   C) 5x^3 - 3x   D) 20x^4 - 6x^2

2. Find d/dx[e^x + ln(x)]

A) e^x + x   B) e^x + 1/x   C) xe^(x-1) + 1/x   D) e^x - 1/x^2

3. Find the derivative of f(x) = x·sin(x) using the product rule.

A) cos(x)   B) sin(x) + x·cos(x)   C) x·cos(x)   D) sin(x) - x·cos(x)

4. Find f'(x) if f(x) = √x = x^(1/2)

A) 1/(2√x)   B) 2√x   C) 1/√x   D) √x/2

5. Use the chain rule: d/dx[sin(3x)]

A) cos(3x)   B) 3cos(3x)   C) -3cos(3x)   D) cos(3)

6. If f(x) = 1/x^2 = x^(-2), then f'(x) = ?

A) -2/x^3   B) 2/x^3   C) -2x^(-1)   D) -1/x^3

7. At what point does f(x) = x^2 - 4x + 1 have a horizontal tangent?

A) x = 0   B) x = 2   C) x = 4   D) x = -2

8. Find d/dx[(2x + 1)^3]

A) 3(2x + 1)^2   B) 6(2x + 1)^2   C) 2(2x + 1)^2   D) 6x(2x + 1)^2

9. If f'(x) = 0 and f''(x) < 0 at x = a, then f has:

A) Local minimum at x = a   B) Local maximum at x = a   C) Inflection point at x = a   D) Cannot determine

10. The slope of y = x^3 at x = -1 is:

A) -3   B) -1   C) 1   D) 3

Click to reveal answers
  1. A) 20x^3 - 6x - Power rule, constant disappears
  2. B) e^x + 1/x - d/dx[e^x] = e^x, d/dx[ln x] = 1/x
  3. B) sin(x) + x·cos(x) - Product rule: sin(x)·1 + x·cos(x)
  4. A) 1/(2√x) - Power rule: (1/2)x^(-1/2) = 1/(2√x)
  5. B) 3cos(3x) - Chain rule: cos(3x)·3
  6. A) -2/x^3 - Power rule: -2x^(-3) = -2/x^3
  7. B) x = 2 - f'(x) = 2x - 4 = 0 when x = 2
  8. B) 6(2x + 1)^2 - Chain rule: 3(2x + 1)^2 · 2
  9. B) Local maximum at x = a - Second derivative test
  10. D) 3 - f'(x) = 3x^2, f'(-1) = 3(1) = 3

Check Your Understanding

1. Explain the connection between the derivative and the slope of a tangent line.

Show answer

The derivative f'(a) gives the slope of the tangent line to the curve y = f(x) at the point where x = a. The tangent line touches the curve at exactly one point (locally) and represents the best linear approximation to the curve at that point. The derivative is defined as the limit of secant line slopes as the two points get infinitely close together.

2. When do you use the product rule versus the chain rule?

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Use the product rule when you have two functions multiplied together: f(x)·g(x). Use the chain rule when you have functions composed (nested): f(g(x)). For example: x·sin(x) needs product rule (multiplication), but sin(x^2) needs chain rule (composition - sin of x^2). Sometimes you need both, like with x·sin(x^2).

3. What does it mean when f'(x) = 0? What additional information do you need?

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When f'(x) = 0, the function has a horizontal tangent at that point - a critical point. This could be a local maximum, local minimum, or neither (inflection point). To determine which, use the second derivative test: if f''(x) > 0, it's a minimum; if f''(x) < 0, it's a maximum; if f''(x) = 0, the test is inconclusive and you need the first derivative test.

4. Why is d/dx[e^x] = e^x significant?

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The function e^x is the only function (other than 0) that equals its own derivative. This makes it incredibly important in calculus and differential equations. It represents natural exponential growth where the rate of change is proportional to the current value - modeling phenomena like compound interest, population growth, and radioactive decay. The number e (approximately 2.718) is defined to make this property work.

🚀 Next Steps

  • Review any concepts that felt challenging
  • Move on to the next lesson when ready
  • Return to practice problems periodically for review