Differentiation: Definition and Fundamental Properties

The slope between two points is change in output divided by change in input. A secant line uses two distinct points. A tangent slope is obtained as those points approach one another, when that limiting slope exists. A derivative has output-units per input-unit.

The difference quotient measures average rate of change over a short interval. Its limit as the interval shrinks defines the derivative. Graphically, secant slopes approach a tangent slope when the derivative exists. A derivative has output-units per input-unit.

The derivative at a point is a number, while the derivative function assigns such rates across its domain. Equivalent limit definitions can use an increment h or an input approaching a fixed value. Recognize both rather than relying on a memorized visual pattern.

Differentiability at an interior point implies continuity there. The converse fails: a continuous function can have a corner, cusp, or vertical tangent. At a corner, the one-sided finite slopes differ. At a vertical tangent, the ordinary finite derivative does not exist.

For a piecewise function to be differentiable at a join, first ensure the values connect continuously, then match the appropriate one-sided derivatives. Matching formulas for slopes while leaving a jump in values does not make the original function differentiable.

The derivative of a constant is zero, and a constant multiplier carries through differentiation. Sums and differences differentiate term by term. The power rule gives the derivative of xⁿ wherever the expression and rule are valid. Do not differentiate a constant base exponential as though its exponent were a fixed power.

Fundamental derivatives include d/dx(e^x) = e^x, d/dx(ln x) = 1x on x > 0, d/dx(sin x) = cos x, and d/dx(cos x) = −sin x with radians. Tangent differentiates to sec²x on its domain.

The derivative of a product is not the product of derivatives. Both factors can change, so the product rule adds two contributions. For a quotient, use a consistent numerator order and square the denominator. A sign error often arises from reversing the quotient-rule subtraction.

Algebraic simplification can sometimes make a derivative easier, but preserve domain restrictions. Rewriting a quotient as a sum of powers may avoid unnecessary work; canceling a factor does not remove an original hole.

A tangent line at x = a passes through (a,f(a)) with slope f′(a). Its equation provides a local linear approximation. It need not intersect the curve only once, and it can cross the curve at the tangency point.

When values are tabulated, a nearby difference quotient can estimate the derivative. A centered interval can be useful when data exist on both sides, but the estimate is not automatically exact. Include units and avoid claiming more precision than the data support.

For f(x)=x², the difference quotient at x is [(x+h)²−x²]h. Expanding gives (2xh+h²)h=2x+h for h≠0. As h approaches zero, the expression approaches 2x. This illustrates why the derivative is a function: each input x has its own local slope. Evaluating that derivative at x=3 gives a number, 6.

The limit definition also explains why differentiability can fail. At a corner, the slopes approached from the two sides disagree. At a vertical tangent, the slope may grow without bound rather than approach a finite number. A discontinuity prevents differentiability because differentiability implies continuity. The reverse implication fails: a graph can be continuous and still have a corner.

A sum can be differentiated term by term, but the derivative of a product is not the product of derivatives. For uv, use u′v+uv′. For uv, use (u′v−uv′)v² where v is nonzero. Writing u, v, u′, and v′ on separate lines can prevent sign and factor errors before substitution.

Simplify first when it removes unnecessary rules. For x²x on x≠0, differentiating the simplified x gives 1 on that domain. Keep the original domain restriction; algebraic simplification does not restore excluded inputs. Conversely, expanding a large product may create more work than applying the product rule. The useful question is which form makes the derivative and its domain easiest to see.

At a differentiable input a, the tangent line is L(x)=f(a)+f′(a)(x−a). The first term anchors the line at the correct point; the second uses the local rate to estimate nearby change. A tangent line can intersect the curve elsewhere and can cross the curve at the point of tangency. It need not remain entirely on one side globally.

For an estimate, choose a nearby input whose function value and derivative are easy to calculate. The approximation is most defensible close to that input. A small input change can still create a noticeable error when curvature is large. Later, concavity helps determine whether a tangent-line estimate lies above or below the true function value.

The difference quotient (f(a+h)−f(a))h is an average rate over a small interval. Its limit as h approaches zero, if it exists, is f′(a). The numerator measures output change and the denominator input change, so derivative units are output units per input unit. The function value f(a), the derivative f′(a), and the nearby change f(a+h)−f(a) answer different questions.

The equivalent quotient (f(x)−f(a))(x−a) approaches the same derivative as x approaches a. Match a given limit to its function and base point before evaluating it. A quotient such as [sin(2+h)−sin 2]h represents the derivative of sine at 2, not at zero.

Differentiability requires compatible finite one-sided derivative behavior. Corners, cusps, vertical tangents, and discontinuities need careful analysis. Continuity is necessary but not sufficient: an absolute-value graph is continuous at its corner but has different one-sided slopes there.

A sum differentiates term by term; a constant factor remains a factor. The power rule applies to powers where the function is defined and differentiable. Rewrite radicals or reciprocals as powers when that simplifies the work, but retain domain restrictions. Algebra before differentiation can reduce unnecessary product or quotient rules.

The product rule has two terms because both factors may change. The quotient rule must retain the denominator squared and the correct subtraction order. Check a result with a simple input or by rewriting the original expression when possible; sign mistakes often become visible through a second representation.

Derivatives of sine and cosine use radian measure. Exponential functions preserve their basic form, while a general base introduces a logarithmic factor. A derivative formula is not a substitute for checking where the original function exists. A simplified derivative may appear defined at an input excluded from the original function.

Where f rises, f′ is positive; where f falls, f′ is negative. A high value of f does not imply a positive derivative. A horizontal tangent gives f′=0 when the derivative exists, but that point need not be a maximum or minimum.

The magnitude of f′ describes local steepness. If a function increases while flattening, its derivative is positive but decreasing. Later, the sign of f″ will describe how f′ changes. Keep these statements separate: increasing value and increasing slope are different properties.

For tabular estimates, use values on both sides of the target when an appropriate centered difference is available. Report an estimate, with units, rather than claiming an exact derivative from finitely many observations. The interval width and smoothness affect accuracy.

After differentiating, check structure before arithmetic. A constant function must have zero derivative. A polynomial of positive degree drops by one degree. A derivative of a position function must have velocity units. These checks do not prove every coefficient is right, but they can quickly expose a missing factor or an inappropriate operation.

When possible, rewrite the original expression and differentiate again. For example, x²x simplifies to x for x≠0, so its derivative is 1 on that domain. The quotient rule must agree. The simplified derivative does not make the original function defined at zero. This is a useful way to separate algebraic equivalence from domain equivalence.

For a numerical derivative estimate, compare nearby secant slopes from both sides. Consistent estimates suggest a derivative; rapidly diverging or incompatible slopes suggest a feature requiring further analysis. A graphing calculator’s smooth display can conceal a corner or hole. Use the definition when the question concerns whether a derivative exists.

Inside each piece’s interval, use the rule for that piece. At the join, begin with continuity: if function values do not approach the assigned value, differentiability fails. Then compare one-sided derivatives. Equal symbolic derivatives away from the join do not compensate for a jump in height.

Suppose the left rule is x² and the right rule is ax+b, meeting at x=1. Continuity requires a+b=1. Matching slopes requires a=2, giving b=−1. Both conditions are needed. Solving only the slope condition leaves a family of lines, most of which do not meet the parabola.

The joining point may also be an endpoint of the overall domain. Read whether the problem asks for a two-sided derivative or a one-sided rate. State the convention rather than assuming a derivative exists where only one side is available.