The Power Rule and other Rules for Differentiation Mr. Miehl

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Presentation transcript:

The Power Rule and other Rules for Differentiation Mr. Miehl

Rules for Differentiation Taking the derivative by using the definition is a lot of work. Perhaps there is an easy way to find the derivative.

Objective  To differentiate functions using the power rule, constant rule, constant multiple rule, and sum and difference rules.

The Derivative is …  Used to find the “slope” of a function at a point.  Used to find the “slope of the tangent line” to the graph of a function at a point.  Used to find the “instantaneous rate of change” of a function at a point.  Computed by finding the limit of the difference quotient as ∆x approaches 0. (Limit Definition)

Notations for the Derivative of a Function “f prime of x” “y prime” “the derivative of y with respect to x” is a verb. “Take the derivative with respect to x…” is a noun.

Rules for Differentiation  Differentiation is the process of computing the derivative of a function. You may be asked to:  Differentiate.  Derive.  Find the derivative of…

Video Clip from Calculus-Help.com The Power Rule

Rules for Differentiation  Working with the definition of the derivative is important because it helps you really understand what the derivative means.

The Power Rule

The Constant Rule  The derivative of a constant function is zero.

The Constant Multiple Rule  The derivative of a constant times a function is equal to the constant times the derivative of the function.

The Sum and Difference Rules The derivative of a sum is the sum of the derivatives. The derivative of a difference is the difference of the derivatives.

Constant Rule  Find the derivative of:

Power Rule  Differentiate:

Constant Multiple Rule  Find the derivative of:

Constant Multiple Rule  Find the derivative of:

Constant Multiple Rule  Find the derivative of:

Rewriting Before Differentiating FunctionRewriteDifferentiateSimplify

FunctionRewriteDifferentiateSimplify

FunctionRewriteDifferentiateSimplify

FunctionRewriteDifferentiateSimplify

Sum & Difference Rules  Differentiate:

Conclusion  Notations for the derivative:  The derivative of a constant is zero.  To find the derivative of f (x) = x N 1.Pull a copy of the exponent out in front of the term. 2.Subtract one from the exponent.