IXL - Find values of derivatives using limits (Calculus practice)

Find the derivative of

(

)

Write your answer as an integer or a fraction. Simplify any fractions.

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question

Find the derivative of

(

)

Write your answer as an integer or a fraction. Simplify any fractions.

key idea

The derivative of

(

)

is:

′

(

)

lim

→0

(

)

–

(

)

The derivative

′

(

)

is both

the instantaneous rate of change of
f
(
x
)
at
x
=
a
,
and
the slope of the tangent line to the curve
y
=
f
(
x
)
at
x
=
a
.

When

(

)

is a linear function, the derivative is always equal to the slope.

solution

The graph of

(

)

is a horizontal line.

Since all horizontal lines have a slope of 0, the derivative is 0.

You can also find the derivative of

(

)

using the limit definition.

′

(

)

lim

→0

(

)

–

(

)

You cannot simply plug in

to evaluate the limit, because doing so makes the denominator 0. You must first simplify the numerator and cancel the

in the denominator.

′

(

)

lim

→0

(

)

–

(

)

lim

→0

–

lim

→0

Cancel

lim

→0

Cancel

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