Calculus Guide

Derivatives Explained

Learn what derivatives mean, how they measure instantaneous change, and how to differentiate functions using the major derivative rules.

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Instantaneous Change

What Does a Derivative Mean?

A derivative measures how quickly one quantity changes with respect to another.

Geometrically, the derivative gives the slope of the tangent line to a graph at a particular point.

In applied problems, the derivative often represents an instantaneous rate of change.

Derivative

\[ f'(x) \]

\(f'(x)\) tells us how the output of \(f(x)\) is changing at each input value \(x\).

Big Idea

A Derivative Is a Rate of Change

If \(y=f(x)\), then \(f'(x)\) describes how rapidly \(y\) changes as \(x\) changes.

A positive derivative means the function is increasing locally, while a negative derivative means the function is decreasing locally.

From Algebra to Calculus

Average Rate vs. Instantaneous Rate

Before calculus, slope describes average change over an interval. Derivatives let us shrink that interval down to a single instant.

Average Rate of Change

\[ \frac{ f(b)-f(a) }{ b-a } \]

This gives the slope of a secant line through two points on the graph.

Instantaneous Rate of Change

\[ f'(a) \]

This gives the slope of the tangent line at one point.

Example

Average Velocity

Suppose position is given by

\[ s(t)=t^2. \]

From \(t=2\) to \(t=4\), the average velocity is

\[ \frac{ s(4)-s(2) }{ 4-2 } = \frac{ 16-4 }{ 2 } = 6. \]

The derivative will let us find velocity at a single instant instead of over an interval.

Several Equivalent Forms

Derivative Notation

Calculus uses several common ways to write a derivative. They all describe the same underlying idea.

Prime Notation

\[ f'(x) \]

Common when working with a named function \(f\).

Leibniz Notation

\[ \frac{dy}{dx} \]

Emphasizes change in \(y\) relative to change in \(x\).

Operator Notation

\[ \frac{d}{dx} \left[f(x)\right] \]

Reads as “differentiate \(f(x)\) with respect to \(x\).”

At a Point

\[ f'(a) \]

The derivative evaluated at the specific input \(x=a\).

Where Derivatives Come From

The Limit Definition of the Derivative

The derivative is built directly from the limit of average rates of change over smaller and smaller intervals.

Difference Quotient

\[ \boxed{ f'(x) = \lim_{h\to0} \frac{ f(x+h)-f(x) }{ h } } \]
Start Two nearby points
Compute Secant slope
Let \(h\to0\)
Result Tangent slope

First Principles

Finding a Derivative from the Definition

Example

Find the Derivative of \(f(x)=x^2\)

Start with

\[ f'(x) = \lim_{h\to0} \frac{ f(x+h)-f(x) }{ h }. \]

Substitute \(f(x)=x^2\):

\[ f'(x) = \lim_{h\to0} \frac{ (x+h)^2-x^2 }{ h }. \]

Expand:

\[ (x+h)^2 = x^2+2xh+h^2. \] \[ f'(x) = \lim_{h\to0} \frac{ x^2+2xh+h^2-x^2 }{ h }. \]

Simplify:

\[ = \lim_{h\to0} \frac{ 2xh+h^2 }{ h }. \] \[ = \lim_{h\to0} \left( 2x+h \right). \]

Now let \(h\to0\):

\[ f'(x)=2x. \]
\[ \boxed{ \frac{d}{dx}(x^2)=2x } \]

Important

The Derivative Is Itself a Function

Since \(f'(x)=2x\), the slope of \(f(x)=x^2\) changes depending on the value of \(x\).

Core Rules

Basic Derivative Rules

Once the meaning of the derivative is understood, most derivatives are found using a small collection of rules.

Constant Rule

\[ \frac{d}{dx}(c)=0 \]

A constant does not change.

Power Rule

\[ \frac{d}{dx} \left(x^n\right) = nx^{n-1} \]

Bring down the exponent and subtract one.

Constant Multiple

\[ \frac{d}{dx} \left[ cf(x) \right] = cf'(x) \]

Constants stay in front.

Sum Rule

\[ \frac{d}{dx} \left[ f(x)+g(x) \right] = f'(x)+g'(x) \]

Differentiate each term.

Difference Rule

\[ \frac{d}{dx} \left[ f(x)-g(x) \right] = f'(x)-g'(x) \]

Differentiate term by term.

The Workhorse Rule

The Power Rule

The power rule is one of the most frequently used derivative rules.

\[ \boxed{ \frac{d}{dx} \left( x^n \right) = nx^{n-1} } \]

Example

\[ \frac{d}{dx}(x^5) \] \[ = 5x^4 \]

Example

\[ \frac{d}{dx}(x^{-2}) \] \[ = -2x^{-3} \]

Example

\[ \frac{d}{dx}(\sqrt{x}) \]

Rewrite first:

\[ \sqrt{x} = x^{1/2} \] \[ = \frac12x^{-1/2} \]

Useful Habit

Rewrite radicals and variables in denominators using exponents before applying the power rule.

Combine the Basics

Differentiating Polynomials

Example

Differentiate

\[ f(x) = 4x^5-3x^3+7x-9. \]

Differentiate each term:

\[ f'(x) = 4(5x^4) - 3(3x^2) + 7 - 0. \] \[ f'(x) = 20x^4-9x^2+7. \]
\[ \boxed{ f'(x)=20x^4-9x^2+7 } \]

Products of Functions

The Product Rule

When two functions are multiplied, you cannot simply differentiate each function and multiply the results.

\[ \boxed{ \frac{d}{dx} \left[ f(x)g(x) \right] = f'(x)g(x) + f(x)g'(x) } \]

Memory Pattern

First derivative × second
+
first × second derivative

Example

Differentiate

\[ y = x^2(x^3+1). \]

Let

\[ f(x)=x^2, \qquad g(x)=x^3+1. \]

Then

\[ f'(x)=2x, \qquad g'(x)=3x^2. \]

Apply the product rule:

\[ y' = (2x)(x^3+1) + x^2(3x^2). \] \[ = 2x^4+2x+3x^4. \] \[ = 5x^4+2x. \]
\[ \boxed{ y'=5x^4+2x } \]

Ratios of Functions

The Quotient Rule

\[ \boxed{ \frac{d}{dx} \left[ \frac{f(x)}{g(x)} \right] = \frac{ f'(x)g(x) - f(x)g'(x) }{ [g(x)]^2 } } \]

Memory Pattern

Bottom times derivative of top minus top times derivative of bottom, all over bottom squared.

Example

Differentiate

\[ y = \frac{x^2+1}{x}. \]

Let

\[ f(x)=x^2+1, \qquad g(x)=x. \] \[ f'(x)=2x, \qquad g'(x)=1. \]

Apply the quotient rule:

\[ y' = \frac{ (2x)(x) - (x^2+1)(1) }{ x^2 }. \] \[ = \frac{ 2x^2-x^2-1 }{ x^2 }. \] \[ = \frac{x^2-1}{x^2}. \]
\[ \boxed{ y' = \frac{x^2-1}{x^2} } \]

Before Using the Quotient Rule

Check whether the expression can be simplified or rewritten using negative exponents first. Sometimes the power rule is easier.

Functions Inside Functions

The Chain Rule

The chain rule is used when one function is composed inside another.

\[ \boxed{ \frac{d}{dx} f(g(x)) = f'(g(x)) g'(x) } \]
Outer \[ (\square)^5 \]
Inner \[ 3x^2+1 \]

Example

Differentiate

\[ y = (3x^2+1)^5. \]

Differentiate the outer function while keeping the inside unchanged:

\[ 5(3x^2+1)^4. \]

Then multiply by the derivative of the inside:

\[ \frac{d}{dx}(3x^2+1) = 6x. \]

Therefore,

\[ y' = 5(3x^2+1)^4(6x). \] \[ y' = 30x(3x^2+1)^4. \]
\[ \boxed{ y' = 30x(3x^2+1)^4 } \]

Common Mistake

Do Not Forget the Inner Derivative

Differentiating \((3x^2+1)^5\) as merely \(5(3x^2+1)^4\) is incomplete. The factor \(6x\) must also appear.

Essential Formulas

Basic Trigonometric Derivatives

Sine

\[ \boxed{ \frac{d}{dx} (\sin x) = \cos x } \]

Cosine

\[ \boxed{ \frac{d}{dx} (\cos x) = -\sin x } \]

Tangent

\[ \boxed{ \frac{d}{dx} (\tan x) = \sec^2 x } \]

Cotangent

\[ \boxed{ \frac{d}{dx} (\cot x) = -\csc^2 x } \]

Secant

\[ \boxed{ \frac{d}{dx} (\sec x) = \sec x\tan x } \]

Cosecant

\[ \boxed{ \frac{d}{dx} (\csc x) = -\csc x\cot x } \]

Important

These derivative formulas assume angles are measured in radians.

Combine the Rules

Trig Functions with the Chain Rule

Example

Differentiate

\[ y = \sin(4x^2). \]

The outer function is sine:

\[ \frac{d}{dx} \sin(u) = \cos(u). \]

So the outer derivative gives

\[ \cos(4x^2). \]

The derivative of the inside is

\[ \frac{d}{dx}(4x^2) = 8x. \]

Multiply:

\[ y' = 8x\cos(4x^2). \]
\[ \boxed{ y' = 8x\cos(4x^2) } \]

Geometric Meaning

Derivatives and Tangent Lines

The derivative at a point gives the slope of the tangent line there.

Tangent Line at \(x=a\)

The slope is

\[ m=f'(a). \]

The point on the curve is

\[ (a,f(a)). \]

Use point-slope form:

\[ \boxed{ y-f(a) = f'(a)(x-a) } \]

Example

Tangent Line to \(f(x)=x^2\) at \(x=2\)

First find the derivative:

\[ f'(x)=2x. \]

Find the slope at \(x=2\):

\[ f'(2)=4. \]

Find the point:

\[ f(2)=4. \]

So the tangent line passes through \((2,4)\) with slope \(4\).

\[ y-4 = 4(x-2). \] \[ y = 4x-4. \]
\[ \boxed{ y=4x-4 } \]

An Important Application

Derivatives and Motion

Position

\[ s(t) \]

Describes where an object is.

Velocity

\[ v(t) = s'(t) \]

The rate of change of position.

Acceleration

\[ a(t) = v'(t) = s''(t) \]

The rate of change of velocity.

Differentiate Again

Higher-Order Derivatives

A derivative can itself be differentiated.

First Derivative

\[ f'(x) \]

Second Derivative

\[ f''(x) \]

Third Derivative

\[ f'''(x) \]

\(n\)th Derivative

\[ f^{(n)}(x) \]

Important Relationship

Differentiability vs. Continuity

Differentiability is a stronger condition than continuity.

If \(f\) Is Differentiable

Then \(f\) must be continuous.

But

Continuity does not guarantee differentiability.

Corner

The left-hand and right-hand slopes do not agree.

Cusp

Slopes become unbounded in opposite directions.

Vertical Tangent

The slope becomes infinite or undefined.

Discontinuity

A function cannot be differentiable where it is discontinuous.

Classic Example

\(f(x)=|x|\)

The graph of \(y=|x|\) is continuous at \(x=0\), but it has a sharp corner there.

The left-hand slope is

\[ -1 \]

while the right-hand slope is

\[ 1. \]

Since the slopes do not agree, \(f'(0)\) does not exist.

Watch Out

Common Derivative Mistakes

1

Forgetting to Reduce the Exponent

For the power rule, multiply by the old exponent and then subtract one from the exponent.

2

Differentiating a Constant Incorrectly

The derivative of any constant is zero.

3

Multiplying Derivatives in the Product Rule

In general,

\[ (fg)' \neq f'g'. \]

Use \(f'g+fg'\).

4

Reversing the Quotient Rule

The numerator must preserve the correct subtraction order.

5

Forgetting the Chain Rule

When a function is nested inside another function, multiply by the derivative of the inside.

6

Forgetting the Negative on Cosine

Remember:

\[ \frac{d}{dx} (\cos x) = -\sin x. \]
7

Confusing \(f'(a)\) with \(f(a)\)

\(f(a)\) is the function value. \(f'(a)\) is the slope or rate of change at that point.

8

Assuming Continuity Means Differentiability

A continuous graph can still have a corner, cusp, or vertical tangent.

Problem-Solving Roadmap

Quick Derivative Strategy

1

Simplify First

Rewrite When Helpful

Rewrite radicals and denominators as powers when that makes differentiation easier.

2

Identify the Structure

What Kind of Function Is It?

Look for powers, products, quotients, trig functions, and compositions.

3

Choose the Rule

Apply the Appropriate Formula

Use the power, product, quotient, chain, or trig derivative rule.

4

Work from Outside In

Watch for the Chain Rule

If one function is inside another, differentiate the outside and multiply by the derivative of the inside.

5

Finish

Simplify the Derivative

Combine like terms and simplify the final expression when appropriate.

Keep This Handy

Derivatives Quick Reference

Limit Definition

\[ f'(x) = \lim_{h\to0} \frac{ f(x+h)-f(x) }{ h } \]

Constant

\[ \frac{d}{dx}(c)=0 \]

Power

\[ \frac{d}{dx}(x^n) = nx^{n-1} \]

Product

\[ (fg)' = f'g+fg' \]

Quotient

\[ \left( \frac{f}{g} \right)' = \frac{ f'g-fg' }{ g^2 } \]

Chain

\[ \frac{d}{dx} f(g(x)) = f'(g(x))g'(x) \]

Sine

\[ (\sin x)' = \cos x \]

Cosine

\[ (\cos x)' = -\sin x \]

Tangent

\[ (\tan x)' = \sec^2x \]

Tangent Line

\[ y-f(a) = f'(a)(x-a) \]

Before You Practice

Derivatives Checklist

  1. Understand the derivative as an instantaneous rate of change.
  2. Connect derivatives with the slope of a tangent line.
  3. Recognize common derivative notation such as \(f'(x)\) and \(\frac{dy}{dx}\).
  4. Understand the limit definition of the derivative.
  5. Know the constant rule and power rule.
  6. Differentiate polynomials term by term.
  7. Use the product rule for products of functions.
  8. Use the quotient rule for ratios of functions.
  9. Recognize compositions and use the chain rule.
  10. Know the basic trig derivative formulas.
  11. Use \(f'(a)\) as the slope of a tangent line.
  12. Understand velocity as the derivative of position and acceleration as the derivative of velocity.
  13. Know that differentiability implies continuity.
  14. Recognize corners, cusps, vertical tangents, and discontinuities as possible reasons a derivative may fail to exist.

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