Calculus 3.3

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3.3 Rules for Differentiation
The Coke Ovens, named because they resemble
brick beehive ovens that were used to
manufacture coke from coal for steel making.
Colorado National Monument
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If the derivative of a function is its slope,
then for a constant function, the derivative must
be zero.
example
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If we find derivatives with the difference
quotient
We observe a pattern

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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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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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Product Rule
We can use the definition of derivative to find a
formula for the derivative of a product.
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Product Rule
We can use the definition of derivative to find a
formula for the derivative of a product.
This would work
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Product Rule
We can use the definition of derivative to find a
formula for the derivative of a product.
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Product Rule
We can use the definition of derivative to find a
formula for the derivative of a product.
We are going to subtract and add the same
expression to the limit
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Using the distributive property
The limit of a product the product of the
limits
Evaluating the limits
Substituting u and v, we get a formula for the
derivative of a product
We are going to use this order to be consistent
with the quotient rule (next) and with the
derivative of cross products (next year.)
Many calculus books (including ours) give this
formula with the terms in a different order.
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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
Once again we can use the definition of
derivative to find a formula.
Clearing the complex fraction
Again we are going to subtract and add the same
expression
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Factoring each side of the numerator and
factoring the denominator
The limit of a product the product of the
limits
Evaluating the limits
Substituting u and v, we get the formula for the
derivative of a quotient
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Quotient Rule
or
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Higher Order Derivatives
(y double prime)
We will learn later what these higher order
derivatives are used for.
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