Derivative as a Limit

Learn derivative definition, derivative of a linear rule.

Lesson 1 of 32 parts Practice Flashcards

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Derivative Definition

Objective: Explain derivative definition, apply the stated method, and check its conditions.

The derivative at a is limₕ→0[f(a+h)−f(a)]/h when the finite limit exists. Algebraic cancellation is performed for h≠0 before taking the limit. A derivative describes local change; existence implies continuity at the point but not conversely. Never substitute h=0 into the original quotient before simplifying.

Formulas

$f′(a)=limₕ→0 [f(a+h)−f(a)]/h$

Worked examples

Try itUse the derivative definition for f(x)=6x² to find f′(7).Show answer

Step 1: Expand [6(7+h)²−294]/h. Step 2: For h≠0, simplify to 84+6h. Step 3: Take h→0 to get 84.

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Derivative Definition

Objective: Explain derivative definition, apply the stated method, and check its conditions.

Formulas

$f′(a)=limₕ→0 [f(a+h)−f(a)]/h$

Worked examples

Try itUse the derivative definition for f(x)=6x² to find f′(7).Show answer

Step 1: Expand [6(7+h)²−294]/h. Step 2: For h≠0, simplify to 84+6h. Step 3: Take h→0 to get 84.

Derivative of a Linear Rule

Objective: Explain derivative of a linear rule, apply the stated method, and check its conditions.

For f(x)=mx+b, the difference quotient is m for every nonzero step. Therefore f′(x)=m at every input. The constant b affects height but not the rate of change; shifting a line vertically leaves its slope unchanged. The derivative of a constant term is zero.

Formulas

$d(mx+b)/dx=m$

Worked examples

Try itUse a difference quotient to find the derivative of f(x)=6x+7.Show answer

Step 1: Compute [f(x+h)−f(x)]/h. Step 2: The numerator is 6h; divide by h≠0. Step 3: The limit is 6.

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