IXL - Find tangent lines using implicit differentiation (Calculus practice)

Find the slope of the tangent line to the graph of

–

(

)

ref_doc_title.

Learn with an example

question

Find the slope of the tangent line to the graph of

(

–

)

key idea

A function

of a variable

may be given in one of two ways: explicitly or

implicitly.

An explicit function

of a variable

is defined by an equation of the form

(

)

An implicit function

of a variable

satisfies an equation between

and

in which

is not isolated.

The derivative of a function provides a rule to find the slope of the tangent line to the graph of that function.

To differentiate an implicit function

with respect to

Differentiate both sides of the equation.
Solve for
y
′
.

To differentiate

use the following formula:

(

)

′

–

′

solution

The slope of the tangent line to a graph at a point is the value of

′

at that point.

The equation

defines

as an implicit function of

since

is not isolated. To find

′

first differentiate both sides of the equation.

(

)

′

(

)

′

(

)

′

(

)

′

(

)

′

Next, plug in

and

–

Then, solve for

′

(

–

)

′

So, the slope of the tangent line is

(You would get the same answer if you instead first solved for

′

in terms of

and

and then plugged in

and

–

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Not feeling ready yet? These can help:

I.1Find derivatives using implicit differentiation

F.7Find equations of tangent lines using limits

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