MATH1851 Chap.5 Derivatives, Tangents and Differentiability
Derivatives, Tangents and Differentiability
A derivative is a limit of slopes
Average rate of change is the slope of a secant through two curve points. Move the second point toward the first and the secant slopes may approach a finite limit. That value is the derivative at the fixed input, the tangent slope and the instantaneous rate of change.
The difference quotient retains this geometry even when the calculation becomes algebraic.
Tangent lines need two pieces of evidence
The derivative supplies a slope, while the original function supplies the point on the curve. Substitute both into point-slope form.
A derivative value alone does not answer a tangent-line question, and a line with the right slope can still be vertically misplaced if the curve point is not checked.
Graph reading works in both directions
Positive derivative values correspond to rising portions of the original graph, negative values to falling portions and zeros to horizontal tangent candidates.
A zero does not automatically mark a maximum or minimum. The derivative sign on both sides determines whether the original graph changes from rising to falling, falling to rising or keeps the same direction.
Differentiability is stricter than continuity
Differentiability at a point guarantees continuity there, but continuity does not guarantee differentiability.
Corners have unequal finite one-sided slopes, cusps have opposite unbounded slopes and vertical tangents have same-direction unbounded slopes. A discontinuity blocks differentiability immediately.
Diagnosing the shape means describing what the one-sided difference quotients do, not merely calling the graph unsmooth.
Keep value, slope and line distinct
The function gives the point, the derivative gives the slope, and point-slope form combines them into a tangent equation.
Checking these three roles separately prevents a derivative value from being reported as a coordinate or a correct slope from being attached to the wrong point.
What this chapter covers
- 01
Secant slopes converging to a tangent
- 02
Difference quotient definitions
- 03
Point-slope equations for tangents
- 04
Positive, negative and zero derivative regions
- 05
Differentiability on intervals and endpoints
- 06
Discontinuity, corners, cusps and vertical tangents
Worked example · free
Build a tangent line from the definition
- 4Write and simplify the difference quotient with a nonzero increment.
- 2Take the limit to obtain the tangent slope.
- 2Evaluate the curve point and write the line.
Key terms
- Secant Slope
- The average rate of change between two curve points separated by a nonzero input gap.
- Tangent Slope
- The limiting secant slope at one point when the difference quotient approaches a finite value.
- Difference Quotient
- The ratio of change in function output to change in input used to define a derivative.
- Derivative Function
- A function assigning the tangent slope or instantaneous rate to each differentiable input.
- Corner
- A continuous point with unequal finite one-sided slopes, so no derivative exists there.
- Vertical Tangent
- A tangent with unbounded slope, so the derivative is not finite under the course definition.
Derivatives, Tangents and Differentiability FAQ
Why does substituting the increment as zero fail?
The raw difference quotient has the increment in its denominator, so immediate substitution creates zero over zero. Simplify the numerator for nonzero nearby increments, cancel the common factor legally and only then take the limit as the increment approaches zero.
How do I sketch a derivative from a function graph?
Mark where the original graph rises, falls and has horizontal tangents. Draw derivative values above the axis on rising intervals, below on falling intervals and at zero at horizontal tangent candidates. Then adjust magnitude to reflect steepness and mark points where no derivative exists.
Does continuity imply differentiability?
No. Differentiability implies continuity, but the converse fails. The absolute-value graph is continuous at its corner while its left slope and right slope disagree. Cusps and vertical tangents provide further continuous points where no finite derivative exists.
Exam move
Alternate three representations: formula, curve and derivative graph. For definition questions, write the quotient, simplify for a nonzero increment, take the limit and then construct the tangent line. For graph questions, mark derivative sign before estimating magnitude. Keep a failure-shape card that distinguishes discontinuity, corner, cusp and vertical tangent by their one-sided slopes.
On piecewise joins, test continuity first and slope agreement second. Verify every tangent line by substituting the original point into the final equation.
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