Derivatives

Calculus

Definition

The Derivative of is

A function is differentiable exists. It is Differentiable on it is differentiable

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Techniques

Chain Rule

Proof

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Continuity

If is Differentiable at then is Continuous at . However the converse is not necessarily true, such as which is continuous at but not differentiable at Proof

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Fermat’s Theorem

If has a local maximum or minimum at then The converse is not true if it is not necessarily a maximum or minimum INSERT PROOF A Critical Number is a number in the domain of such that or doesn’t exist

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Rolle’s Theorem

If is Continuous on , Differentiable on , then INSERT PROOF

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Mean Value Theorem

If is Continuous on , Differentiable on then

This means that at some point the slop of the tangent is equal to the slope of the secant line which represents average rate of change hence the name

INSERT PROOF

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Concavity

1. is on (convex downward/concave upward)
2. is concave on (convex upward/convex downward)

A point on is an inflection point if is continuous at and it changes direction of concavity.