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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

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0.56mins

Understanding second derivative notation of Implicit Differentiation

Find `y''`

of `xy + y^2 = 2`

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4.04mins

Finding the Second Derivative by Implicit Differentiation

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1.53mins

Finding the derivative of Inverse Cos

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1.56mins

Finding the derivative of Inverse Tan arctan

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3.11mins

Higher Order Derivative ex1

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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
```

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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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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