Basic Differentiation Rules

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Presentation transcript:

Basic Differentiation Rules

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.

Examples Find derivatives for the following functions: Find the equation of the line tangent to the graph of at the point

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

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.

Theorem. Example. Find all values of x where the line tangent to the graph of has slope –1.

The Derivative As a Rate of Change Slope.

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

The Derivative an Instantaneous Rate of Change

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?