Introduction to Implicit Differentiation
Finding instantaneous rate of change on non functions.
ex Find \frac{dy}{dx}
for x^2 + y^2 =1
.
ex Find \frac{dy}{dx}
for x^3 +y^3 = 6xy
then find the tangent at x = 3
.
Implicit Differentiation ex1 Finding equation of tangent when x and y are given
Implicit Differentiation ex2 with Trig Ratios
Implicit Differentiation ex3 Finding Horizontal or Vertical Tangents
Find y''
of xy + y^2 = 2
Determine \frac{dy}{dx}
for the following in terms of x
and y
, using implicit differentiation:
\displaystyle
x^2 + y^2 = 36
Determine \frac{dy}{dx}
for the following in terms of x
and y
, using implicit differentiation:
\displaystyle
15y^2 = 2x^3
Determine \frac{dy}{dx}
for the following in terms of x
and y
, using implicit differentiation:
\displaystyle
3xy^2 + y^3 = 8
Determine \frac{dy}{dx}
for the following in terms of x
and y
, using implicit differentiation:
\displaystyle
9x^2-16y^2 = - 144
Determine \frac{dy}{dx}
for the following in terms of x
and y
, using implicit differentiation:
\displaystyle
\frac{x^2}{16} + \frac{3y^2}{13} = 1
Determine \frac{dy}{dx}
for the following in terms of x
and y
, using implicit differentiation:
\displaystyle
x^2 + y^2 + 5y = 10
For the relation, determine the equation of the tangent at the given point.
\displaystyle
x^2 + y^2 = 13, (2, -3))
For the relation, determine the equation of the tangent at the given point.
\displaystyle
x^2 + 4y^2 = 100, (-8, 3)
For the relation, determine the equation of the tangent at the given point.
\displaystyle
\frac{x^2}{25} - \frac{y^2}{36} = - 1, (5\sqrt{3}, -12)
For the relation, determine the equation of the tangent at the given point.
\displaystyle
\frac{x^2}{81} - \frac{5y^2}{162} = 1, (-11, -4)
At what point is the tangent to the curve x+ y^2 = 1
parallel to the line x + 2y = 0
?
The equation 5x^2 - 6xy + 5y^2 = 16
represents an ellipse.
a. Detrmine \frac{dy}{dx}
at (1, -1)
.
b. Determine two points on the ellipse at which the tangent is horizontal.
Determine the slope of the tangent to the ellipse 5x^2 + y^2 = 21
at the point A(-2, -1)
.
Determine the equation of the normal to the curve x^3 + y^3 - 3xy = 17
at the point (2, 3).
Determine the equation of the normal to the curve y^2 = \frac{x^3}{2- x}
at the point (1, -1).
Determine \frac{dy}{dx}
(x + y)^3 = 12x
Determine \frac{dy}{dx}
\sqrt{x + y} - 2x = 1
The equation 4x^2y - 3y = x^3
implicitly defines y
as a function of x
.
a. Use implicit differentiation to determine \frac{dy}{dx}
.
b. Write y
as an explicit function of x
, and compute \frac{dy}{dx}
directly.
Show that \frac{dy}{dx} = \frac{y}{x}
for the relation \sqrt{\frac{x}{y}} + \sqrt{\frac{y}{x}} = 10, x
, y \neq 0
.
Determine the equations of the lines that are tangent to the ellipse \displaystyle
x^2 + 4y^2 = 16
and also pass through the point (4, 6)
.
The angle between two intersecting curves is defined as the angle between their tangents at the point of intersection. If this angle is 90°, the two curves are said to be orthogonal at this point.
Prove that the curves defined by x^2-y^2 = k
and xy = p
intersect orthogonally for all values of the constants k
and p
. Illustrate your proof with a sketch.
Let l
be any tangent to the curve \sqrt{x} + \sqrt{y} = \sqrt{k}
, where k
is a constant. Show that the sum of the intercepts of l
is k
.
Two circles of radius 3\sqrt{2}
are tangent to the graph y^2 = 4x
at the point (1, 2)
. Determine the equations of these two circles.