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Basic Differentiation Rules
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Derivative Rules Theorem. [The Constant Rule] If k is a real number such that for all x in some open interval I, then for all Theorem. [The Power Rule] Let r be a rational number, and let Then for all values of x where this expression is defined.
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Examples Find derivatives for the following functions:
Find the equation of the line tangent to the graph of at the point
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More Derivative Rules Theorem [The Constant Multiple Rule] Let k represent a real number, and let f be a differentiable function. Then the function kf is also differentiable and Example. Find the derivative of
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Theorem [The Sum and Difference Rules] Let f and g be differentiable functions. Then
Example. Find the derivative of each function. Note. This theorem generalizes to any finite sum or difference.
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Theorem. Example. Find all values of x where the line tangent to the graph of has slope –1.
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The Derivative As a Rate of Change
Slope.
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Velocity. Let be a function giving the position of a point moving on a number line at time t.
The derivative gives the instantaneous velocity at time
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The Derivative an Instantaneous Rate of Change
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Example. A stone dropped from a bridge falls in t seconds
Example. A stone dropped from a bridge falls in t seconds. Find the velocity after 3 seconds. If a river flows 256 feet below the bridge, with what velocity does the rock enter the water?
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