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3.3 Rules for Differentiation PowerPoint PPT Presentation


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Photo by Vickie Kelly, 2003. Greg Kelly, Hanford High School, Richland, Washington. 3.3 Rules for Differentiation. Colorado National Monument. The derivative of a constant is zero. If the derivative of a function is its slope, then for a constant function, the derivative must be zero.

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3.3 Rules for Differentiation

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Photo by Vickie Kelly, 2003

Greg Kelly, Hanford High School, Richland, Washington

3.3 Rules for Differentiation

Colorado National Monument


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The derivative of a constant is zero.

If the derivative of a function is its slope, then for a constant function, the derivative must be zero.

example:


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(Pascal’s Triangle)

If we find derivatives with the difference quotient:

We observe a pattern:


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We observe a pattern:

power rule

examples:


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constant multiple rule:

examples:

When we used the difference quotient, we observed that since the limit had no effect on a constant coefficient, that the constant could be factored to the outside.


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constant multiple rule:

sum and difference rules:

(Each term is treated separately)


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Horizontal tangents occur when slope = zero.

Example:

Find the horizontal tangents of:

Plugging the x values into the original equation, we get:

(The function is even, so we only get two horizontal tangents.)


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First derivative (slope) is zero at:


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product rule:

Notice that this is not just the product of two derivatives.

This is sometimes memorized as:


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quotient rule:

or


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is the first derivative of y with respect to x.

is the second derivative.

is the third derivative.

is the fourth derivative.

Higher Order Derivatives:

(y double prime)

We will learn later what these higher order derivatives are used for.

p


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