10.1 Implicit Differentiation
Chapter
Chapter 10
Section
10.1
Lectures 11 Videos

Introduction to Implicit Differentiation

Finding instantaneous rate of change on non functions.

ex Find \frac{dy}{dx} for x^2 + y^2 =1.

5.03mins
Introduction to Implicit Differentiation

ex Find \frac{dy}{dx} for x^3 +y^3 = 6xy then find the tangent at x = 3.

4.55mins
Implicit Differentiation ex1 when x is given only

Implicit Differentiation ex1 Finding equation of tangent when x and y are given

2.21mins
Implicit Differentiation ex1 Finding equation of tangent when x and y are given

## Understanding second derivative notation of Implicit Differentiation of Leibniz

0.56mins
Understanding second derivative notation of Implicit Differentiation

## Finding the Second Derivative by Implicit Differentiation

Find y'' of xy + y^2 = 2

4.04mins
Finding the Second Derivative by Implicit Differentiation

## Finding the derivative of Inverse Cos arccos

1.53mins
Finding the derivative of Inverse Cos

## Finding the derivative of Inverse Tan arctan

1.56mins
Finding the derivative of Inverse Tan arctan

## Higher Order Derivative ex1

3.11mins
Higher Order Derivative ex1

## Higher Order Derivative ex2

4.21mins
Higher Order Derivative ex2
Solutions 23 Videos

Determine \frac{dy}{dx} for the following in terms of x and y, using implicit differentiation:

 \displaystyle x^2 + y^2 = 36 

0.39mins
Q2a

Determine \frac{dy}{dx} for the following in terms of x and y, using implicit differentiation:

 \displaystyle 15y^2 = 2x^3 

0.34mins
Q2b

Determine \frac{dy}{dx} for the following in terms of x and y, using implicit differentiation:

 \displaystyle 3xy^2 + y^3 = 8 

1.25mins
Q2c

Determine \frac{dy}{dx} for the following in terms of x and y, using implicit differentiation:

 \displaystyle 9x^2-16y^2 = - 144 

0.51mins
Q2d

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 

1.11mins
Q2e

Determine \frac{dy}{dx} for the following in terms of x and y, using implicit differentiation:

 \displaystyle x^2 + y^2 + 5y = 10 

0.34mins
Q2f

For the relation, determine the equation of the tangent at the given point.

 \displaystyle x^2 + y^2 = 13, (2, -3)) 

0.53mins
Q3a

For the relation, determine the equation of the tangent at the given point.

 \displaystyle x^2 + 4y^2 = 100, (-8, 3) 

0.56mins
Q3b

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) 

2.19mins
Q3c

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) 

2.04mins
Q3d

At what point is the tangent to the curve x+ y^2 = 1 parallel to the line x + 2y = 0 ?

1.06mins
Q4

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.

5.06mins
Q5

Determine the slope of the tangent to the ellipse 5x^2 + y^2 = 21 at the point A(-2, -1).

0.55mins
Q6

Determine the equation of the normal to the curve x^3 + y^3 - 3xy = 17 at the point (2, 3).

1.50mins
Q7

Determine the equation of the normal to the curve y^2 = \frac{x^3}{2- x} at the point (1, -1).

1.52mins
Q8

Determine \frac{dy}{dx}

(x + y)^3 = 12x

0.41mins
Q9a

Determine \frac{dy}{dx}

\sqrt{x + y} - 2x = 1

0.56mins
Q9b

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.

4.39mins
Q10

Show that \frac{dy}{dx} = \frac{y}{x} for the relation \sqrt{\frac{x}{y}} + \sqrt{\frac{y}{x}} = 10, x , y \neq 0.

0.00mins
Q12

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).

Q13

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.