Derivatives are one of the fundamental concepts of calculus and find applications in physics, engineering, economics, and all quantitative sciences. Understanding differentiation rules and applying them correctly is essential for studying functions, optimization, and mathematical modeling. In this comprehensive guide, we present the theory, rules, formulas, and numerous step-by-step worked examples.
What Is the Derivative of a Function
The derivative of a function f(x) at a point x₀ represents the instantaneous rate of change of the function at that point. Geometrically, it corresponds to the slope of the tangent line to the graph of the function at the given point.
Formal Definition
The derivative of f at the point x₀ is defined as the limit of the difference quotient:
f'(x₀) = lim [f(x₀ + h) - f(x₀)] / h as h approaches 0
If this limit exists and is finite, f is said to be differentiable at x₀. The derivative function f'(x) is obtained by calculating this limit for a generic point x in the domain.
Notations
The derivative of f(x) can be denoted using several equivalent notations:
| Notation | Author | Meaning |
|---|---|---|
| f'(x) | Lagrange | First derivative of f |
| df/dx | Leibniz | Derivative of f with respect to x |
| Df(x) | Euler | Derivative operator applied to f |
| f with dot above | Newton | Used mainly in physics (derivative with respect to time) |
Geometric Meaning
If we consider the graph of f(x), the difference quotient [f(x₀ + h) - f(x₀)] / h represents the slope of the secant line passing through the points (x₀, f(x₀)) and (x₀ + h, f(x₀ + h)). As h approaches zero, the secant "rotates" until it becomes the tangent line, and its slope is precisely the derivative.
Derivatives of Elementary Functions
Before tackling differentiation rules, it is essential to know the derivatives of basic functions. These should be memorized and represent the "building blocks" for constructing more complex derivatives.
| Function f(x) | Derivative f'(x) | Notes |
|---|---|---|
| c (constant) | 0 | The derivative of a constant is always zero |
| x | 1 | Special case of x^n with n=1 |
| x^n | n · x^(n-1) | For any real n (power rule) |
| √x = x^(1/2) | 1 / (2√x) | Special case of x^n with n=1/2 |
| 1/x = x^(-1) | -1/x² | Special case of x^n with n=-1 |
| e^x | e^x | The exponential equals its own derivative |
| a^x | a^x · ln(a) | For a > 0, a ≠ 1 |
| ln(x) | 1/x | For x > 0 |
| log_a(x) | 1 / (x · ln(a)) | Logarithm base a |
| sin(x) | cos(x) | Angle in radians |
| cos(x) | -sin(x) | Angle in radians |
| tan(x) | 1/cos²(x) = 1 + tan²(x) | Where cos(x) ≠ 0 |
| arcsin(x) | 1 / √(1-x²) | For |x| < 1 |
| arccos(x) | -1 / √(1-x²) | For |x| < 1 |
| arctan(x) | 1 / (1+x²) | For any real x |
Differentiation Rules
Constant Multiple Rule
[c · f(x)]' = c · f'(x)
A multiplicative constant "comes out" of the derivative.
Example: D[5x³] = 5 · 3x² = 15x²
Sum Rule (and Difference Rule)
[f(x) ± g(x)]' = f'(x) ± g'(x)
The derivative of a sum is the sum of the derivatives.
Example: D[x³ + 2x² - 7x + 4] = 3x² + 4x - 7
Product Rule
[f(x) · g(x)]' = f'(x) · g(x) + f(x) · g'(x)
The derivative of the product of two functions is calculated by differentiating the first and multiplying by the second, then adding the first multiplied by the derivative of the second.
Example: D[x² · sin(x)] = 2x · sin(x) + x² · cos(x)
Quotient Rule
[f(x) / g(x)]' = [f'(x) · g(x) - f(x) · g'(x)] / [g(x)]²
Pay attention to the order: "derivative of the numerator times denominator, minus numerator times derivative of the denominator, all divided by the denominator squared."
Example: D[x / (x² + 1)] = [1 · (x² + 1) - x · 2x] / (x² + 1)² = (1 - x²) / (x² + 1)²
Chain Rule (Composite Function)
[f(g(x))]' = f'(g(x)) · g'(x)
The chain rule is the most important and most commonly used rule. You differentiate the outer function evaluated at the inner function, and multiply by the derivative of the inner function.
Example: D[sin(3x²)] = cos(3x²) · 6x
Example: D[e^(x² + 1)] = e^(x² + 1) · 2x
Example: D[ln(x² + 5)] = [1/(x² + 5)] · 2x = 2x/(x² + 5)
Generalized Power Rule
[f(x)^n]' = n · [f(x)]^(n-1) · f'(x)
A combination of the power rule with the chain rule.
Example: D[(3x + 2)³] = 5(3x + 2)² · 3 = 15(3x + 2)²
Step-by-Step Worked Examples
Example 1: Derivative of a Polynomial
Calculate f'(x) for f(x) = 4x⁵ - 3x³ + 2x² - 7x + 1
We apply the sum rule and the power rule term by term:
- D[4x⁵] = 20x⁴
- D[-3x³] = -9x²
- D[2x²] = 4x
- D[-7x] = -7
- D[1] = 0
f'(x) = 20x⁴ - 9x² + 4x - 7
Example 2: Product Rule
Calculate the derivative of f(x) = x² · e^x
Let u = x² and v = e^x. Then u' = 2x and v' = e^x.
f'(x) = u'v + uv' = 2x · e^x + x² · e^x = e^x(2x + x²) = e^x · x(x + 2)
Example 3: Quotient Rule
Calculate the derivative of f(x) = (x² - 1) / (x² + 1)
Numerator: f = x² - 1, f' = 2x. Denominator: g = x² + 1, g' = 2x.
f'(x) = [2x(x² + 1) - (x² - 1) · 2x] / (x² + 1)²
= [2x³ + 2x - 2x³ + 2x] / (x² + 1)²
= 4x / (x² + 1)²
Example 4: Multiple Chain Rule
Calculate the derivative of f(x) = sin²(3x) = [sin(3x)]²
There are three levels: power, sine, linear function.
- Derivative of the power: 2 · sin(3x)
- Times the derivative of sine: cos(3x)
- Times the derivative of the inner function 3x: 3
f'(x) = 2 · sin(3x) · cos(3x) · 3 = 6 sin(3x) cos(3x) = 3 sin(6x)
(In the last step, we used the identity 2 sin(α) cos(α) = sin(2α))
Example 5: Composite Logarithmic Function
Calculate the derivative of f(x) = ln(√(x² + 1))
First simplify: f(x) = (1/2) · ln(x² + 1)
f'(x) = (1/2) · [2x / (x² + 1)] = x / (x² + 1)
Example 6: Derivative of a Composite Exponential Function
Calculate the derivative of f(x) = e^(sin(x))
By the chain rule:
f'(x) = e^(sin(x)) · D[sin(x)] = e^(sin(x)) · cos(x) = cos(x) · e^(sin(x))
Applications of Derivatives
1. Tangent Line to a Curve
The derivative f'(x₀) provides the slope of the tangent line to the graph of f at the point (x₀, f(x₀)). The equation of the tangent line is:
y - f(x₀) = f'(x₀) · (x - x₀)
Example: tangent line to f(x) = x² at the point x₀ = 3.
- f(3) = 9, f'(x) = 2x, f'(3) = 6
- y - 9 = 6(x - 3) → y = 6x - 9
2. Maxima and Minima (Optimization)
The relative maximum and minimum points of a function are found where the derivative equals zero (stationary points) and changes sign:
- If f'(x) changes from positive to negative: relative maximum
- If f'(x) changes from negative to positive: relative minimum
- If f'(x) does not change sign: inflection point with horizontal tangent
Alternatively, the second derivative test can be used: if f'(x₀) = 0 and f''(x₀) > 0 → minimum; if f''(x₀) < 0 → maximum.
3. Studying Increasing and Decreasing Behavior
The derivative indicates where a function is increasing or decreasing:
- f'(x) > 0: the function is increasing
- f'(x) < 0: the function is decreasing
- f'(x) = 0: the function has a stationary point
4. Velocity and Acceleration in Physics
If s(t) represents the position of an object over time:
- Velocity: v(t) = s'(t) (first derivative of position)
- Acceleration: a(t) = v'(t) = s''(t) (second derivative of position)
Example: if s(t) = 5t² + 3t + 2 (position in meters, time in seconds):
- v(t) = 10t + 3 m/s (velocity)
- a(t) = 10 m/s² (constant acceleration)
5. Rate of Change in Economics
If C(q) is the total cost function as a function of quantity produced q:
- Marginal cost: C'(q) = derivative of total cost, indicates the additional cost of producing one more unit
- Marginal revenue: R'(q) = derivative of total revenue
- Maximum profit: achieved when R'(q) = C'(q)
Higher-Order Derivatives
The derivative of a derivative is called the second derivative, denoted f''(x) or d²f/dx². Derivatives of any order can be calculated:
| Order | Notation | Meaning |
|---|---|---|
| First | f'(x) | Rate of change, slope |
| Second | f''(x) | Concavity, acceleration |
| Third | f'''(x) | Rate of change of acceleration (jerk in physics) |
| n-th | f^(n)(x) | n-th order derivative |
Example: f(x) = x⁴ - 3x³ + x
- f'(x) = 4x³ - 9x² + 1
- f''(x) = 12x² - 18x
- f'''(x) = 24x - 18
- f⁴(x) = 24
- f⁵(x) = 0
Common Errors in Derivative Calculations
Here are the most frequent mistakes to avoid:
| Error | Wrong | Correct |
|---|---|---|
| Forgetting the chain rule | D[sin(3x)] = cos(3x) | D[sin(3x)] = 3cos(3x) |
| Product differentiated as product of derivatives | D[x² · sin(x)] = 2x · cos(x) | D[x² · sin(x)] = 2x sin(x) + x² cos(x) |
| Wrong order in the quotient rule | D[f/g] = (fg' - f'g)/g² | D[f/g] = (f'g - fg')/g² |
| Derivative of e^x with coefficient | D[e^(2x)] = e^(2x) | D[e^(2x)] = 2e^(2x) |
| Sign of cosine | D[cos(x)] = sin(x) | D[cos(x)] = -sin(x) |
Frequently Asked Questions (FAQ)
What is the difference between a derivative and a differential?
The derivative f'(x) is a number (or a function) that represents the instantaneous rate of change. The differential df = f'(x) · dx is a linear approximation of the change in the function. The derivative is the ratio of the differentials: f'(x) = df/dx.
Are all functions differentiable?
No. A function must be continuous to be differentiable, but continuity is not sufficient. For example, f(x) = |x| is continuous at x = 0 but not differentiable (it has a "corner"). There also exist functions that are continuous everywhere but differentiable nowhere (Weierstrass function).
What is the second derivative used for?
The second derivative provides information about the concavity of the function: if f''(x) > 0 the function is concave up (convex), if f''(x) < 0 it is concave down. It is also used to determine the nature of stationary points and to find inflection points (where f''(x) = 0 and changes sign).
How do you differentiate a function raised to a function?
For functions of the form f(x)^g(x), use logarithmic differentiation: set y = f(x)^g(x), take the logarithm of both sides (ln y = g(x) · ln f(x)), and differentiate implicitly. The result is: y' = f(x)^g(x) · [g'(x) · ln f(x) + g(x) · f'(x)/f(x)].
Is the derivative of a sum always the sum of the derivatives?
Yes, the derivative has the linearity property: the derivative of a sum (or difference) of functions is the sum (or difference) of the derivatives. Note: this property does not hold for the product or the quotient.
How do you calculate derivatives of implicit functions?
For functions defined implicitly by an equation F(x, y) = 0, differentiate both sides with respect to x, treating y as a function of x (applying the chain rule every time y appears), and then isolate dy/dx. For example, from x² + y² = 1 we get 2x + 2y · y' = 0, from which y' = -x/y.
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