04 The Derivative

Meaning of the Derivative

The derivative describes change at a single input value. For a function $f$, the derivative at $a$ is defined by

provided the limit exists. The expression inside the limit is the difference quotient. It compares the change in output, $f(a+h)-f(a)$, with the small input change $h$.

Over a complete interval from $x=a$ to $x=b$, the average rate of change is

This is the slope of a secant line. When $b$ moves closer to $a$, the secant line approaches the tangent line, and its slope approaches the instantaneous rate of change, which is the derivative.

For example, if a particle's position is $s(t)$, then

is its average velocity between two times, while $s'(a)$ is its velocity at the exact time $a$.

Takeaway: The derivative is both a limiting slope and an instantaneous rate of change.

Derivative Notation and Motion

Several notations represent the same derivative:

The notation $f'(x)$ describes a new function, while $f'(a)$ means that this derivative function has been evaluated at the particular input $a$.

For position and motion, if $s(t)$ is position, then velocity is

and acceleration is

Thus, the second derivative measures how velocity changes. Units also carry meaning: if position is measured in meters and time in seconds, then velocity has units of meters per second and acceleration has units of meters per second squared.

Takeaway: Notation identifies what is being differentiated, which variable is changing, and whether the result is a function or a value at a particular point.

Tangent Lines and Differentiability

If a function is differentiable at $x=a$, its tangent line at $(a,f(a))$ has slope $f'(a)$. The point-slope equation is

For $f(x)=x^2$, the derivative is $f'(x)=2x$. At $x=3$, the point on the curve is $(3,9)$, and the slope is $f'(3)=6$. Therefore,

which simplifies to

When $x$ is close to $a$, the tangent line gives the local linear approximation

This approximation is useful because a complicated curve and its tangent line have nearly the same values near the point of tangency.

A function is differentiable at a point when its derivative exists there. Differentiability requires a well-defined finite tangent slope. A discontinuity, hole, sharp corner, cusp, or vertical tangent can cause the derivative to fail to exist. Differentiability implies continuity, but continuity does not imply differentiability. For instance, $f(x)=|x|$ is continuous at $0$, but its left-hand slope is $-1$ and its right-hand slope is $1$, so it has no derivative at $0$.

Takeaway: A derivative determines the tangent-line slope, and differentiability is a stronger condition than continuity.

Core Differentiation Rules

The basic differentiation rules make symbolic calculation efficient.

• Constant rule:

  $$\frac{d}{dx}[c]=0.$$

• Power rule:

  $$\frac{d}{dx}[x^n]=nx^{n-1}.$$

• Constant multiple rule:

  $$\frac{d}{dx}[cf(x)]=cf'(x).$$

• Sum and difference rules:

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

For a polynomial, differentiate each term separately. For example,

has derivative

The constant term disappears because its derivative is zero.

For products, use the product rule:

For quotients, use the quotient rule:

For nested expressions, use the chain rule:

For example,

The factor $6x$ comes from differentiating the inner function $3x^2+1$.

Takeaway: First identify the structure of an expression—sum, product, quotient, or composition—then select the corresponding rule.

Exponential and Logarithmic Derivatives

The exponential and logarithmic rules follow distinctive patterns. For the natural exponential function,

With a composite exponent,

For a general exponential function with $a>0$ and $a\ne 1$,

and therefore

The natural logarithm satisfies

for $x>0$. Applying the chain rule gives

where the logarithm's input must be positive. For example,

For a logarithm with base $a$,

Takeaway: Exponential derivatives preserve the exponential factor, while logarithmic derivatives produce a reciprocal factor and require attention to domain.

Trigonometric Derivatives

The standard trigonometric derivative formulas use radians as the angle measure:

If the trigonometric function contains a composite argument, multiply by the derivative of that argument. For example,

The inner function is $4x^3-1$, whose derivative is $12x^2$. The chain rule supplies this factor.

Takeaway: Memorize the six basic trigonometric derivatives, use radians, and apply the chain rule to nontrivial arguments.

Interpreting and Applying Derivatives

A derivative can be interpreted directly from its sign and magnitude.

• If $f'(x)>0$, the function is increasing locally.

• If $f'(x)<0$, the function is decreasing locally.

• If $f'(x)=0$, the tangent line is horizontal. This may correspond to a local maximum, a local minimum, or neither.

• A large value of $|f'(x)|$ indicates a steep graph and a rapid local rate of change.

The units of a derivative are the units of the output divided by the units of the input. If $C(q)$ is the cost in dollars of producing $q$ items, then $C'(q)$ has units of dollars per item. It approximates the additional cost associated with producing one more item near the production level $q$.

A reliable workflow is:

1. Identify the function and the variable with respect to which you are differentiating.

2. Determine the expression's structure.

3. Apply the appropriate rule, including the chain rule for compositions.

4. Simplify the derivative.

5. Evaluate at a specified input if a numerical rate or tangent slope is required.

6. Check the domain and interpret the units and sign.

Takeaway: Differentiation connects algebraic formulas with geometric slopes, physical motion, and practical rates of change.