Derivatives: A Comprehensive Guide
Estimated reading time, 7 minutes 📝
Understanding Derivatives
Before diving into the table, let's clarify what derivatives are. In the realm of calculus, a derivative represents the rate at which a function changes with respect to its input. It's a fundamental concept with applications across various fields, including physics, economics, and engineering.
The Derivative Table
Function  Derivative 

Constant (c)  0 
x  1 
x^n  n*x^(n1) 
√x  1/(2√x) 
1/x  1/x^2 
e^x  e^x 
a^x  (ln a) * a^x 
ln x  1/x 
log_a x  1/(x * ln a) 
sin x  cos x 
cos x  sin x 
tan x  sec^2 x 
cot x  csc^2 x 
sec x  sec x * tan x 
csc x  csc x * cot x 
Using the Derivative Table
To find the derivative of a function, identify the corresponding function in the table and apply the given formula. For instance, the derivative of x^3 is 3x^(31) = 3x^2.
Key Points to Remember
 Constant Rule: The derivative of a constant is always zero.
 Power Rule: To find the derivative of x^n, multiply by the exponent n and decrease the exponent by 1.
 Exponential and Logarithmic Functions: Have specific derivative formulas.
 Trigonometric Functions: Have their own set of derivatives.
Applications of Derivatives
Derivatives have a wide range of applications:
 Slope of a Curve: Finding the slope of a tangent line at a specific point on a curve.
 Optimization: Determining maximum and minimum values of a function.
 Related Rates: Analyzing how rates of change of different quantities are related.
 Physics: Calculating velocity and acceleration.
Derivatives: Beyond the Basics
Derivative Rules
While the table provides the fundamental derivatives, several rules are crucial for handling more complex functions:
Sum and Difference Rule
 d/dx [f(x) ± g(x)] = f'(x) ± g'(x)
 The derivative of a sum/difference is the sum/difference of the derivatives.
Product Rule
 d/dx [f(x) * g(x)] = f'(x) * g(x) + f(x) * g'(x)
 The derivative of a product involves the derivative of the first function times the second, plus the first function times the derivative of the second.
Quotient Rule
 d/dx [f(x) / g(x)] = [f'(x) * g(x)  f(x) * g'(x)] / [g(x)]^2
 The derivative of a quotient is a bit more complex, involving the derivative of the numerator times the denominator minus the numerator times the derivative of the denominator, all divided by the denominator squared.
Chain Rule
 d/dx [f(g(x))] = f'(g(x)) * g'(x)
 Used for composite functions, the derivative of the outer function evaluated at the inner function, multiplied by the derivative of the inner function.
Examples
Let's apply these rules to a few examples:

Example 1: Find the derivative of f(x) = x^2 * sin(x)
 Using the product rule: f'(x) = (2x) * sin(x) + x^2 * cos(x)

Example 2: Find the derivative of g(x) = (x^3 + 2x) / (x  1)
 Using the quotient rule: g'(x) = [(3x^2 + 2)(x  1)  (x^3 + 2x)(1)] / (x  1)^2

Example 3: Find the derivative of h(x) = sin(x^2)
 Using the chain rule: h'(x) = cos(x^2) * 2x
Visualizing Derivatives
As mentioned earlier, the derivative represents the slope of a tangent line to a curve at a specific point. This visual interpretation reinforces the concept of the derivative as a rate of change.
Applications of Derivatives
Beyond the fundamental concepts, derivatives find applications in:
 Physics: Velocity, acceleration, force, work, and energy.
 Economics: Marginal cost, revenue, and profit, elasticity.
 Engineering: Optimization problems, rates of change in various systems.
 Geometry: Finding the equation of a tangent line, curve sketching.
Implicit Differentiation and HigherOrder Derivatives
Implicit Differentiation
So far, we've focused on finding derivatives of functions explicitly defined in terms of x. However, sometimes, equations implicitly define relationships between x and y. In these cases, we use implicit differentiation.
Steps:
 Differentiate both sides of the equation with respect to x.
 Treat y as a function of x, and apply the chain rule when differentiating y terms.
 Solve the resulting equation for dy/dx.
Example: Find dy/dx for the equation x^2 + y^2 = 25.
 Differentiate both sides: 2x + 2y * dy/dx = 0
 Solve for dy/dx: dy/dx = x/y
HigherOrder Derivatives
The derivative of a derivative is called a higherorder derivative. The second derivative is denoted as f''(x) or d^2y/dx^2, the third as f'''(x) or d^3y/dx^3, and so on.
Example: Find the second derivative of f(x) = x^3.
 First derivative: f'(x) = 3x^2
 Second derivative: f''(x) = 6x
Applications of HigherOrder Derivatives
 Physics: Acceleration is the second derivative of position with respect to time.
 Concavity: The second derivative determines the concavity of a curve (whether it's concave up or concave down).
 Optimization: Higherorder derivatives can be used to classify critical points as local maxima, minima, or inflection points.
Exercises
 Find dy/dx for the equation x^3 + y^3 = 6xy.
 Find the second derivative of f(x) = sin(x).
Derivatives: Applications and Beyond
Optimization Problems
One of the most practical applications of derivatives is in optimization problems. These involve finding maximum or minimum values of a function.
Steps:
 Identify the function to be optimized.
 Find the critical points by setting the first derivative equal to zero and solving for x.
 Use the second derivative test to determine if the critical points correspond to maxima, minima, or inflection points.
Example: Find the dimensions of a rectangle with maximum area that can be inscribed in a semicircle of radius r.
Related Rates
Related rates problems involve finding the rate of change of one quantity in terms of the rate of change of another quantity.
Steps:
 Identify the given rates and the rate to be found.
 Write down the equation relating the quantities involved.
 Differentiate both sides of the equation with respect to time (implicit differentiation).
 Substitute known values and solve for the desired rate.
Example: A spherical balloon is being inflated at a rate of 100 cm^3/s. How fast is the radius increasing when the radius is 5 cm?
Applications in Physics
Derivatives are fundamental in physics. Some examples include:
 Velocity: The derivative of position with respect to time.
 Acceleration: The derivative of velocity with respect to time.
 Force: Related to the derivative of potential energy.
Economic Applications
Derivatives are used extensively in economics:
 Marginal cost/revenue/profit: These are the rates of change of cost, revenue, and profit with respect to the quantity produced.
 Elasticity: Measures the responsiveness of demand to changes in price.
Additional Topics
 Logarithmic Differentiation: Used for functions that involve products, quotients, and powers of x.
 Exponential Growth and Decay: Derivatives play a crucial role in modeling these phenomena.
 Partial Derivatives: Used for functions of multiple variables.
Exercises
 Find two positive numbers whose product is 100 and whose sum is a minimum.
 A ladder 10 ft long rests against a vertical wall. If the bottom of the ladder slides away from the wall at a rate of 1 ft/s, how fast is the top of the ladder sliding down the wall when the bottom of the ladder is 6 ft from the wall?
Derivatives Terminology
Term  Definition 

Derivative  The rate of change of a function with respect to its input. 
Differentiation  The process of finding a derivative. 
Tangent line  A line that touches a curve at exactly one point and has the same slope as the curve at that point. 
Instantaneous rate of change  The rate of change of a function at a specific point. 
Average rate of change  The rate of change of a function over an interval. 
Secant line  A line that intersects a curve at two points. 
Limit  The value a function approaches as its input approaches a specific value. 
Continuity  A function is continuous if it can be drawn without lifting the pen. 
Differentiability  A function is differentiable if its derivative exists at every point in its domain. 
Power rule  For differentiating functions of the form x^n. 
Product rule  For differentiating the product of two functions. 
Quotient rule  For differentiating the quotient of two functions. 
Chain rule  For differentiating composite functions. 
Implicit differentiation  For differentiating implicitly defined functions. 
Logarithmic differentiation  For differentiating functions involving products, quotients, and powers. 
Second derivative  The derivative of the first derivative. 
Concavity  The direction in which a curve bends. 
Inflection point  A point where the concavity of a curve changes. 
Critical point  A point where the first derivative is zero or undefined. 
Local maximum/minimum  A point where a function has a maximum or minimum value within a specific interval. 
Global maximum/minimum  The highest or lowest point of a function over its entire domain. 
Optimization  Finding maximum or minimum values of a function. 
Related rates  Finding the rate of change of one quantity in terms of the rate of change of another. 
Marginal analysis  Analyzing the rate of change of cost, revenue, or profit. 
Velocity and acceleration  The first and second derivatives of position with respect to time. 
Curve sketching  Using derivatives to analyze the shape of a graph. 
Partial derivative  The derivative of a multivariable function with respect to one variable, holding the others constant. 
Gradient  A vector that points in the direction of the greatest rate of increase of a function. 
Directional derivative  The rate of change of a function in a specific direction. 
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