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2.7. DERIVATIVES OF FUNCTIONS GIVEN IMPLICITLY
• While
dy
dx may now involve both the variables x and y,
dy
dx still measures the slope of
the tangent line to the curve, and thus this derivative may be used to decide when the
tangent line is horizontal (
dy
dx = 0) or vertical (
dy
dx is undefined), or to find the equation
of the tangent line at a particular point on the curve.
Exercises
1. Consider the curve given by the equation 2y 3 + y 2 − y 5 = x 4 − 2x 3 + x 2 . Find all
points at which the tangent line to the curve is horizontal or vertical. Be sure to use a
graphing utility to plot this implicit curve and to visually check the results of algebraic
reasoning that you use to determine where the tangent lines are horizontal and vertical.
2. For the curve given by the equation sin(x + y) + cos(x − y) = 1, find the equation of
the tangent line to the curve at the point (
π
2 ,
π
2 ).
3. Implicit differentiation enables us a different perspective from which to see why the
rule
d
dx [a x ] = a x ln(a) holds, if we assume that
d
dx [ln(x)] =
1
x . This exercise leads you
through the key steps to do so.
(a) Let y = a x . Rewrite this equation using the natural logarithm function to write
x in terms of y (and the constant a).
(b) Differentiate both sides of the equation you found in (a) with respect to x,
keeping in mind that y is implicitly a function of x.
(c) Solve the equation you found in (b) for
dy
dx , and then use the definition of y to
write
dy
dx solely in terms of x. What have you found?
2.7. DERIVATIVES OF FUNCTIONS GIVEN IMPLICITLY
• While
dy
dx may now involve both the variables x and y,
dy
dx still measures the slope of
the tangent line to the curve, and thus this derivative may be used to decide when the
tangent line is horizontal (
dy
dx = 0) or vertical (
dy
dx is undefined), or to find the equation
of the tangent line at a particular point on the curve.
Exercises
1. Consider the curve given by the equation 2y 3 + y 2 − y 5 = x 4 − 2x 3 + x 2 . Find all
points at which the tangent line to the curve is horizontal or vertical. Be sure to use a
graphing utility to plot this implicit curve and to visually check the results of algebraic
reasoning that you use to determine where the tangent lines are horizontal and vertical.
2. For the curve given by the equation sin(x + y) + cos(x − y) = 1, find the equation of
the tangent line to the curve at the point (
π
2 ,
π
2 ).
3. Implicit differentiation enables us a different perspective from which to see why the
rule
d
dx [a x ] = a x ln(a) holds, if we assume that
d
dx [ln(x)] =
1
x . This exercise leads you
through the key steps to do so.
(a) Let y = a x . Rewrite this equation using the natural logarithm function to write
x in terms of y (and the constant a).
(b) Differentiate both sides of the equation you found in (a) with respect to x,
keeping in mind that y is implicitly a function of x.
(c) Solve the equation you found in (b) for
dy
dx , and then use the definition of y to
write
dy
dx solely in terms of x. What have you found?