Derivative

The rate of change of a function with respect to a variable

Definition

The derivative of a function measures the rate at which the function's output changes as its input changes. Geometrically, it represents the slope of the tangent line to the curve of the function at a given point.

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

\[f'(a) = \lim_{h \to 0} \frac{f(a+h) - f(a)}{h}\]

Alternatively, using the alternative notation:

\[\frac{df}{dx}\bigg|_{x=a} = \lim_{x \to a} \frac{f(x) - f(a)}{x - a}\]

If this limit exists, we say that f is differentiable at x = a.

Interpretations

Geometric Interpretation

The derivative at a point represents the slope of the tangent line to the curve at that point. If the derivative is positive, the function is increasing; if negative, the function is decreasing.

Physical Interpretation

In physics, if s(t) represents the position of an object at time t, then:

  • s'(t) = v(t) is the velocity
  • s''(t) = a(t) is the acceleration

Economic Interpretation

In economics, the derivative represents marginal change:

  • Marginal cost: the cost of producing one additional unit
  • Marginal revenue: the revenue from selling one additional unit

Differentiation Rules

Basic Rules

Constant Rule: \[\frac{d}{dx}[c] = 0\]
Power Rule: \[\frac{d}{dx}[x^n] = nx^{n-1}\]
Constant Multiple Rule: \[\frac{d}{dx}[cf(x)] = c \cdot f'(x)\]
Sum/Difference Rule: \[\frac{d}{dx}[f(x) \pm g(x)] = f'(x) \pm g'(x)\]

Product and Quotient Rules

Product Rule: \[\frac{d}{dx}[f(x) \cdot g(x)] = f'(x)g(x) + f(x)g'(x)\]
Quotient Rule: \[\frac{d}{dx}\left[\frac{f(x)}{g(x)}\right] = \frac{f'(x)g(x) - f(x)g'(x)}{[g(x)]^2}\]

Chain Rule

Chain Rule: \[\frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x)\]

Or in Leibniz notation: \[\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}\]

Common Derivatives

Trigonometric Functions

\[\frac{d}{dx}[\sin x] = \cos x\]
\[\frac{d}{dx}[\cos x] = -\sin x\]
\[\frac{d}{dx}[\tan x] = \sec^2 x\]

Exponential and Logarithmic Functions

\[\frac{d}{dx}[e^x] = e^x\]
\[\frac{d}{dx}[a^x] = a^x \ln a\]
\[\frac{d}{dx}[\ln x] = \frac{1}{x}\]
\[\frac{d}{dx}[\log_a x] = \frac{1}{x \ln a}\]

Inverse Trigonometric Functions

\[\frac{d}{dx}[\arcsin x] = \frac{1}{\sqrt{1-x^2}}\]
\[\frac{d}{dx}[\arccos x] = -\frac{1}{\sqrt{1-x^2}}\]
\[\frac{d}{dx}[\arctan x] = \frac{1}{1+x^2}\]

Examples

Example 1: Find the derivative of f(x) = x³ + 2x² - 5x + 3.

Solution:

Using the power rule for each term:

\[f'(x) = 3x² + 4x - 5\]

Example 2: Find the derivative of f(x) = x² · sin(x).

Solution:

Using the product rule where u = x² and v = sin(x):

\[f'(x) = 2x \cdot \sin(x) + x² \cdot \cos(x)\]

Example 3: Find the derivative of f(x) = (3x² + 1)⁵.

Solution:

Using the chain rule:

\[f'(x) = 5(3x² + 1)⁴ · 6x = 30x(3x² + 1)⁴\]

Example 4: Find the derivative of f(x) = e^(2x).

Solution:

Using the chain rule:

\[f'(x) = e^(2x) · 2 = 2e^(2x)\]

Example 5: Find the derivative of f(x) = ln(x² + 1).

Solution:

Using the chain rule:

\[f'(x) = \frac{1}{x² + 1} · 2x = \frac{2x}{x² + 1}\]

Higher-Order Derivatives

The derivative of a derivative is called the second derivative, denoted as:

\[f''(x) = \frac{d²f}{dx²} = \frac{d}{dx}\left[\frac{df}{dx}\right]\]

Similarly, the nth derivative is:

\[f^{(n)}(x) = \frac{d^n f}{dx^n}\]

Interpretation of Second Derivative

  • If f''(x) > 0, the function is concave up (like a cup)
  • If f''(x) < 0, the function is concave down (like a cap)
  • Points where f''(x) = 0 or is undefined may be inflection points

Applications

Optimization

Derivatives are used to find maximum and minimum values of functions:

  • Find critical points by solving f'(x) = 0
  • Use the first or second derivative test to classify them
  • Applications: maximizing profit, minimizing cost, optimizing designs

Related Rates

Derivatives help solve problems where multiple quantities change with respect to time:

  • How fast is the water level rising?
  • How fast is the shadow moving?
  • How fast are two cars approaching each other?

Curve Sketching

Derivatives provide information about the shape of a curve:

  • Where the function is increasing or decreasing
  • Location of local maxima and minima
  • Points of inflection and concavity
  • Asymptotic behavior

Notation

Several notations are commonly used for derivatives:

Notation Name Usage
f'(x) Lagrange Most common in calculus
df/dx Leibniz Emphasizes variable relationship
Df(x) Euler Operator notation
ẏ Newton Physics, time derivatives