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Graph Analyzis with Derivatives



The Derivative at a Point
Tangent Lines
Increasing and Decreasing Functions
Critical Points
The First Derivative Test
The Second Derivative Test
Concavity
Inflection Points
Curve Sketching
Optimization
Related Rates
The Full Derivative Toolkit at a Glance



Reading Functions Through Their Derivatives


A function's graph carries information about direction, curvature, turning points, and long-run behavior. The derivative extracts this information systematically. Where the derivative is positive, the function rises. Where it is negative, the function falls. Where it equals zero, something potentially interesting happens—a peak, a valley, or a momentary pause.

The second derivative adds depth. It determines whether the graph bends upward or downward, distinguishing between a function that accelerates and one that decelerates. Together, the first and second derivatives provide a complete toolkit for analyzing the shape of any differentiable function.

This page develops the full framework: from the derivative at a single point through tangent lines, monotonicity, extrema, concavity, inflection points, curve sketching, optimization, and related rates.

Key Terms

First Derivative

Tangent Line— line through (a,f(a))(a, f(a)) with slope f′(a)f'(a)
Critical Point— where f′=0f' = 0 or f′f' is undefined
Local Extremum— local maximum or minimum at a critical point
Monotonic Function— f′>0f' > 0 means increasing, f′<0f' < 0 means decreasing

Second Derivative

Concavity— f′′>0f'' > 0 concave up, f′′<0f'' < 0 concave down
Inflection Point— where concavity reverses

See All Calculus Definitions →


Tangent Lines


The tangent line to ff at x=ax = a passes through (a,f(a))(a, f(a)) with slope f′(a)f'(a). Its equation in point-slope form is:

Tangent Line Equation
y−f(a)=f′(a)(x−a)y=f(a)+f′(a)(x−a)y - f(a) = f'(a)(x - a) \qquad y = f(a) + f'(a)(x - a)
Learn more about this formula: Tangent Line Equation →

This line is the best linear approximation to ff near x=ax = a. For values of xx close to aa, the tangent line and the curve are nearly indistinguishable. This is the geometric foundation of linear approximation.

The normal line at the same point is perpendicular to the tangent. If f′(a)≠0f'(a) \neq 0, its slope is −1/f′(a)-1/f'(a). If f′(a)=0f'(a) = 0, the tangent is horizontal and the normal is vertical.

Normal Line Equation
y−f(a)=−1f′(a)(x−a)y - f(a) = -\frac{1}{f'(a)}(x - a)
Learn more about this formula: Normal Line Equation →


Increasing and Decreasing Functions


The sign of f′f' on an interval determines the direction of ff on that interval.

If f′(x)>0f'(x) > 0 for all xx in an open interval (a,b)(a, b), then ff is strictly increasing on (a,b)(a, b): larger inputs produce larger outputs. If f′(x)<0f'(x) < 0 on (a,b)(a, b), then ff is strictly decreasing: larger inputs produce smaller outputs.

Sign of First Derivative
f′(x)>0  ⟹  f increasing,f′(x)<0  ⟹  f decreasingf'(x) > 0 \implies f \text{ increasing}, \quad f'(x) < 0 \implies f \text{ decreasing}
Learn more about this formula: Sign of First Derivative →


The proof relies on the Mean Value Theorem. For any two points x1<x2x_1 < x_2 in (a,b)(a, b), there exists cc between them with f(x2)−f(x1)=f′(c)(x2−x1)f(x_2) - f(x_1) = f'(c)(x_2 - x_1). If f′(c)>0f'(c) > 0 and x2−x1>0x_2 - x_1 > 0, then f(x2)−f(x1)>0f(x_2) - f(x_1) > 0, confirming ff is increasing.

To find where a function increases or decreases: solve f′(x)=0f'(x) = 0 and identify where f′f' is undefined, then test the sign of f′f' in each resulting interval.

Critical Points


A critical point of ff is a value x=cx = c in the domain of ff where either f′(c)=0f'(c) = 0 or f′(c)f'(c) does not exist.

Critical Point Condition
f′(c)=0orf′(c) undefinedf'(c) = 0 \quad \text{or} \quad f'(c) \text{ undefined}
Learn more about this formula: Critical Point Condition →


Critical points are the only candidates for local extrema. If ff has a local maximum or minimum at cc, then cc must be a critical point. This is Fermat's theorem: a local extremum at an interior point requires the derivative to vanish or fail to exist.

The converse is false. The function f(x)=x3f(x) = x^3 has f′(0)=0f'(0) = 0, so x=0x = 0 is a critical point, but ff has no extremum there—it increases through the origin. A critical point is a candidate that must be tested further.

Finding critical points is the first step in any extremum or optimization problem. The subsequent tests—first derivative test and second derivative test—classify what happens at each candidate.

The First Derivative Test


The first derivative test classifies a critical point cc by examining how f′f' changes sign around it.

If f′f' changes from positive to negative at cc—the function rises then falls—then ff has a local maximum at cc. If f′f' changes from negative to positive—the function falls then rises—then ff has a local minimum at cc. If f′f' does not change sign—positive on both sides or negative on both sides—then cc is neither a maximum nor a minimum.

First Derivative Test
f′(x) changes +→− at c  ⟹  c is a local maxf'(x) \text{ changes } + \to - \text{ at } c \implies c \text{ is a local max}
f′(x) changes −→+ at c  ⟹  c is a local minf'(x) \text{ changes } - \to + \text{ at } c \implies c \text{ is a local min}
Learn more about this formula: First Derivative Test →


The test works even when f′(c)f'(c) does not exist, as long as ff is continuous at cc. It requires checking the sign of f′f' in an interval immediately to the left and immediately to the right of cc. No information about f′′f'' is needed, making this test universally applicable among derivative-based classification methods.

The Second Derivative Test


When f′(c)=0f'(c) = 0 and f′′(c)f''(c) exists, the second derivative test provides a quicker classification.

If f′′(c)>0f''(c) > 0, the graph is concave up at cc, and the horizontal tangent sits at the bottom of a cup: ff has a local minimum at cc. If f′′(c)<0f''(c) < 0, the graph is concave down, and the horizontal tangent sits at the top of a cap: ff has a local maximum at cc.

Second Derivative Test
f′(c)=0,  f′′(c)>0  ⟹  c is a local minf'(c) = 0, \; f''(c) > 0 \implies c \text{ is a local min}
f′(c)=0,  f′′(c)<0  ⟹  c is a local maxf'(c) = 0, \; f''(c) < 0 \implies c \text{ is a local max}
Learn more about this formula: Second Derivative Test →


If f′′(c)=0f''(c) = 0, the test is inconclusive. The point might be an extremum or an inflection point—further analysis is required. In this case, fall back to the first derivative test or examine higher-order derivatives.

The second derivative test is faster than the first derivative test when f′′f'' is easy to compute, but it cannot handle critical points where f′f' does not exist. The table below sets the two tests side by side across their hypotheses, mechanics, and verdicts.
Test When applicable How it works Verdict at c
First derivative test f continuous at c (works even if f'(c) does not exist) examine the sign of f' immediately to the left and right of c + → −: local maximum;  − → +: local minimum;  no sign change: neither
Second derivative test f'(c) = 0 and f''(c) exists evaluate the sign of f''(c) f''(c) > 0: local minimum;  f''(c) < 0: local maximum;  f''(c) = 0: inconclusive

Concavity


Concavity describes how the slope of ff changes, not whether the function rises or falls but how it bends while doing so.

If f′′(x)>0f''(x) > 0 on an interval, the derivative f′f' is increasing there: the slope gets steeper (or less negative). The graph bends upward—concave up. Visually, the curve lies above its tangent lines.

If f′′(x)<0f''(x) < 0 on an interval, the derivative f′f' is decreasing: the slope gets less steep (or more negative). The graph bends downward—concave down. The curve lies below its tangent lines.

Concavity from Second Derivative
f′′(x)>0  ⟹  f concave up,f′′(x)<0  ⟹  f concave downf''(x) > 0 \implies f \text{ concave up}, \quad f''(x) < 0 \implies f \text{ concave down}
Learn more about this formula: Concavity from Second Derivative →


A function can be increasing and concave down simultaneously—rising but decelerating, like a ball thrown upward before reaching its peak. Concavity and direction are independent properties controlled by f′′f'' and f′f' respectively. The four sign combinations of f′f' and f′′f'' produce four qualitatively distinct shapes, collected in the table below alongside the visual picture each one produces.
Sign of f' Sign of f'' What f does on the interval Visual picture
+ + increasing and concave up rising with slope getting steeper
+ − increasing and concave down rising but flattening — like a thrown ball approaching its peak
− + decreasing and concave up falling but flattening — approaching a local minimum from the left
− − decreasing and concave down falling with slope getting more negative

Inflection Points


An inflection point is a point where the concavity of ff changes—from concave up to concave down, or the reverse.

Inflection Point Condition
f′′(c)=0 or undefined, and f′′(x) changes sign at cf''(c) = 0 \text{ or undefined}, \text{ and } f''(x) \text{ changes sign at } c
Learn more about this formula: Inflection Point Condition →


At an inflection point, f′′f'' must either equal zero or fail to exist. However, f′′(c)=0f''(c) = 0 alone does not guarantee an inflection point. The function f(x)=x4f(x) = x^4 has f′′(0)=0f''(0) = 0, but the concavity does not change at x=0x = 0—the graph is concave up on both sides. The sign of f′′f'' must actually switch across the point.

To locate inflection points: find where f′′(x)=0f''(x) = 0 or f′′f'' is undefined, then verify that f′′f'' changes sign. On the graph of f′f', inflection points of ff correspond to local extrema of f′f'—the slope of ff reaches a peak or trough and reverses direction.

Curve Sketching


Curve sketching assembles all derivative information into a complete picture of a function's graph.

The process begins with preliminary analysis: domain, intercepts, symmetry (even, odd, periodic), and asymptotes. Vertical asymptotes arise where the function is undefined; horizontal asymptotes come from limits at infinity.

First derivative analysis determines where ff increases and decreases, and locates all local extrema. Second derivative analysis determines concavity intervals and inflection points. End behavior—the function's direction as x→±∞x \to \pm\infty—frames the overall shape.

Combining these elements produces a qualitative sketch without plotting individual points. The derivative framework reveals structure that pointwise computation alone would miss: a function might appear flat in a table of values while the derivatives expose a subtle extremum nearby.

Optimization


Optimization finds the absolute maximum or minimum value of a function, often on a specified interval or subject to constraints.

On a closed interval [a,b][a, b], the Extreme Value Theorem guarantees that a continuous function attains both an absolute maximum and an absolute minimum. The candidates are the critical points inside (a,b)(a, b) and the endpoints aa and bb. Evaluate ff at each candidate; the largest value is the absolute maximum, the smallest is the absolute minimum.

Extreme Value Theorem
f continuous on [a,b]  ⟹  f attains a max and min on [a,b]f \text{ continuous on } [a, b] \implies f \text{ attains a max and min on } [a, b]
Learn more about this formula: Extreme Value Theorem →


Applied optimization problems translate a real scenario into a function of one variable. The steps are: identify the quantity to optimize, express it as a function of a single variable, determine the feasible domain, find critical points within that domain, and compare values at critical points and endpoints.

The derivative identifies candidates; the context determines which candidate solves the problem. Checking that a critical point is actually a maximum (or minimum) rather than merely a critical point is an essential final step. The table below collects this procedure as a five-step recipe.
Step Action Why it matters
1 identify the quantity to optimize names the objective the derivative will be taken of
2 express it as a function of a single variable constraints are used here to eliminate extra variables
3 determine the feasible domain physical or geometric meaning restricts where the variable can live
4 find the critical points inside that domain candidates for the extremum come from f'(x) = 0 or f'(x) undefined
5 compare values at critical points and at the endpoints on a closed interval, the Extreme Value Theorem guarantees the maximum and minimum lie among these

Related Rates


Related rates problems involve two or more quantities that change simultaneously with respect to a common variable, usually time tt. A known rate of change in one quantity determines an unknown rate of change in another, connected through an equation.

The method uses implicit differentiation with respect to tt. Start with an equation relating the variables—geometric, physical, or algebraic. Differentiate both sides with respect to tt, applying the chain rule to every variable that depends on tt. Substitute known values and known rates, then solve for the unknown rate.

Setting up the relationship correctly is the critical step. The equation must hold at all times, not just at the particular instant in question. Constants can be substituted only after differentiation—replacing a variable with a fixed value before differentiating eliminates its rate of change from the equation.

The Full Derivative Toolkit at a Glance


Every section above introduced a separate way the first or second derivative answers a question about the graph of f — direction, steepness, extrema, concavity, inflection, sketching, optimization, related rates. The table below lays them all out side by side: each tool, the geometric or applied question it settles, and the specific derivative information it needs.
Tool What it reveals Derivative information needed
Value of f'(a) slope and direction of f at the single point (a, f(a)) first derivative evaluated at a
Tangent line best linear approximation to f near x = a:  y − f(a) = f'(a)(x − a) first derivative evaluated at a
Sign of f' intervals where f is increasing or decreasing first derivative across each interval
Critical points candidates for local extrema (must be tested further) solutions of f'(x) = 0 plus points where f' is undefined
First derivative test classifies a critical point as max, min, or neither sign of f' on each side of the critical point
Second derivative test faster classification when f'' is available and nonzero value of f''(c) at a critical point with f'(c) = 0
Sign of f'' intervals of concave up or concave down behavior second derivative across each interval
Inflection points points where concavity reverses f''(x) = 0 or undefined, plus a sign change of f'' across the point
Curve sketching a complete qualitative picture: monotonicity, extrema, concavity, asymptotes f' and f'' combined with domain, intercepts, and end behavior
Optimization absolute maximum or minimum on a domain or under a constraint critical points of the objective function, plus endpoint values
Related rates how an unknown rate of change depends on known rates of related quantities implicit differentiation of a constraint equation with respect to time