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Differentiability

Unit 2: Introduction to Derivatives

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Differentiability

Examples written by:
Kalyan Karamsetty
Image Credit:
Julian Zett

Examples Coming Soon!

Reference

A function must be continuous at a point for it be differentiable at that point; however, just because a function is continuous at a point doesn't necessarily mean it's differentiable at that point. A function is not differentiable at cusps or corners, discontinuities, or vertical asymptotes. Another way to think about differentiability is that the derivative function itself must be continuous at a point for the original function to be differentiable at that point.

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EX 1

Is the function \(\left|x\right|\) differentiable at \(x=0\)?

EX 2

Multiple Choice: Let \(f\) be the function defined by \(f\left(x\right)=\frac{1}{x-2}\) Which of the folowing statements are true?

\[\]

\(\mbox{I.}\) \(f\) is differentiable at \(x=2\)

\(\mbox{II.}\) \(f\) is not continous at \(x=2\)

\(\mbox{III.}\) \(f\) has a vertical asymptote at \(x=2\)

\[\]

\(\mbox{A.}\) \(\mbox{I}\), \(\mbox{II}\), and \(\mbox{III}\)

\(\mbox{B.}\) \(\mbox{I}\) and \(\mbox{II}\)

\(\mbox{C.}\) \(\mbox{II}\) only

\(\mbox{D.}\) \(\mbox{III}\) only

\(\mbox{E.}\) \(\mbox{II}\) and \(\mbox{III}\)

EX 3

Multiple Choice: Let \(f\) be the function defined by \(f\left(x\right)=\sqrt{\left|x+2\right|}\) Which of the following statements are true?

\(\mbox{A.}\) \(f\) is not continuous at \(x=-2\)

\(\mbox{B.}\) \(f\) is continuous and differentiable at \(x=-2\)

\(\mbox{C.}\) \(f\) is not differentiable at \(x=-2\)

\(\mbox{D.}\) \(x=-2\) is a vertical asymptote of the graph \(f\)

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All topics in this unit

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Introduction to Limits

Unit 1: Limits

Introduction to Limits

Unit 1: Limits

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Properties of Limits

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Properties of Limits

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Limits Involving Infinity

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Continuity

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The Derivative

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Differentiability

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Differentiability

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Rules of Differentiation

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Rules of Differentiation

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Derivatives of Trigonometric Functions

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The Chain Rule

Unit 2: Introduction to Derivatives

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Kinematics with Derivatives

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Exponential and Logarithmic Derivatives

Unit 3: More with Derivatives

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Inverse Trigonometric Derivatives

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Implicit Differentiation

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Hyperbolic Functions

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Derivatives of Inverse Functions

Unit 3: More with Derivatives

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Unit 3: More with Derivatives

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Related Rates

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L'Hôpital's Rule

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Critical Points

Unit 4: Analyzing Functions

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Points of Inflection

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Points of Inflection

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Extrema

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Extrema

Unit 4: Analyzing Functions

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Curve Sketching

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The Mean Value Theorem

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Optimization

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Optimization

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Linear Approximation

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Linear Approximation

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Riemann Sums

Unit 5: Integrals

Riemann Sums

Unit 5: Integrals

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Fundamental Theorem of Calculus

Unit 5: Integrals

Fundamental Theorem of Calculus

Unit 5: Integrals

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Antiderivatives

Unit 5: Integrals

Antiderivatives

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Integration by Substitution

Unit 5: Integrals

Integration by Substitution

Unit 5: Integrals

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Definite Integrals

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Definite Integrals

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Area Under Curves

Unit 6: Applications of Integration

Area Under Curves

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Volumes by Cylindrical Shells

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Rectilinear Motion

Unit 6: Applications of Integration

Rectilinear Motion

Unit 6: Applications of Integration

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Kinematics with Integrals

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Kinematics with Integrals

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Average Value of functions

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Unit 6: Applications of Integration

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Differential Equations

Unit 7: Differential Equations

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Slope Fields

Unit 7: Differential Equations

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Unit 7: Differential Equations

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Euler’s Method

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Linear First Order Differential Equations

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