10.1 Implicit Differentiation
Chapter
Chapter 10
Section
10.1
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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.

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5.03mins
Introduction to Implicit Differentiation

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

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4.55mins
Implicit Differentiation ex1 when x is given only

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

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

Implicit Differentiation ex2 with Trig Ratios

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2.44mins
Implicit Differentiation ex2 with Trig Ratios

Implicit Differentiation ex3 Finding Horizontal or Vertical Tangents

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6.55mins
Implicit Differentiation ex3 Finding Horizontal or Vertical Tangents

Understanding second derivative notation of Implicit Differentiation of Leibniz

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0.56mins
Understanding second derivative notation of Implicit Differentiation

Finding the Second Derivative by Implicit Differentiation

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

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4.04mins
Finding the Second Derivative by Implicit Differentiation

Finding the derivative of Inverse Cos arccos

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1.53mins
Finding the derivative of Inverse Cos

Finding the derivative of Inverse Tan arctan

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1.56mins
Finding the derivative of Inverse Tan arctan
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

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0.39mins
Q2a

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

\displaystyle 15y^2 = 2x^3

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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

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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

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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

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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

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0.34mins
Q2f

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

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

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0.53mins
Q3a

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

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

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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)

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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)

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2.04mins
Q3d

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

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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.

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5.06mins
Q5

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

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0.55mins
Q6

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

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1.50mins
Q7

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

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1.52mins
Q8

Determine \frac{dy}{dx}

(x + y)^3 = 12x

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0.41mins
Q9a

Determine \frac{dy}{dx}

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

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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.

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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.

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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).

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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.

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Q14

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.

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Q15

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.

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Q16