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Solutions
27 Videos

Distinguish between natural logarithms and common logarithms.

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0.41mins

Q1

Determine the derivative.

```
\displaystyle
y = \ln(5x + 8)
```

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0.28mins

Q3a

Determine the derivative.

```
\displaystyle
y = \ln(x^2 + 1)
```

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0.22mins

Q3b

Determine the derivative.

```
\displaystyle
v = e^t\ln t
```

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0.59mins

Q3c

Determine the derivative.

```
\displaystyle
v = \ln \sqrt{x + 1}
```

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0.37mins

Q3d

Determine the derivative.

```
\displaystyle
s = \ln (t^3 -2t^2 + 5)
```

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0.30mins

Q3e

Determine the derivative.

```
\displaystyle
w = \ln \sqrt{z^2 + 3z}
```

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1.03mins

Q3f

Differentiate.

```
\displaystyle
f(x) = x \ln x
```

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0.17mins

Q4a

Differentiate.

```
\displaystyle
y = e^{\ln x}
```

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0.21mins

Q4b

Differentiate.

```
\displaystyle
y = e^{x} \ln x
```

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0.27mins

Q4c

Differentiate.

```
\displaystyle
g(x) = \ln(e^{-x} + x e^{-x})
```

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0.43mins

Q4d

Differentiate.

```
\displaystyle
s = \ln(x^3 - 2x^2 + 5)
```

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1.06mins

Q4e

Differentiate.

```
\displaystyle
y = \ln \sqrt{x^2 + 3x}
```

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1.48mins

Q4f

If `g(x) = e^{2x -1}\ln(2x - 1)`

, evaluate `g'(1)`

.

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1.04mins

Q5a

If `f(x) = \ln(\frac{x - 1}{3x + 5})`

, evaluate `f'(5)`

.

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0.55mins

Q5b

Solve the equation for `f'(x) = 0`

:

`f(x) = \ln(x^2 + 1)`

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0.28mins

Q6a

Solve the equation for `f'(x) = 0`

:

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Q6b

Solve the equation for `f'(x) = 0`

:

`f(x) = (x^2 + 1)^{-1}\ln(x^2 + 1)`

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1.04mins

Q6c

Determine the equation of the tangent to the curve defined by `f(x) = \frac{\ln \sqrt[3]{x}}{x}`

at the point where x = 1.

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1.30mins

Q7a

Determine the equation of the tangent to the curve defined by `y = \ln x - 1`

that is parallel to the straight line with equation `3x - 6y - 1 = 0`

.

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1.47mins

Q8

If `f(x) = (x\ln x)^2`

, determine al the points at which the graph of `f(x)`

has a horizontal tangent line.

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2.10mins

Q9a

Determine the equation of the tangent to the curve defined by `y = \ln(1 + e^{-x})`

at the point where `x = 0`

.

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1.10mins

Q10

The velocity, in kilometres per hour, of a car as it begins to slow down is given by the equation `v(t) = 90- 30\ln(3t + 1)`

, where t is in seconds.

What is the velocity of the car as the driver beings to brake?

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0.39mins

Q11a

The velocity, in kilometres per hour, of a car as it begins to slow down is given by the equation `v(t) = 90- 30\ln(3t + 1)`

, where t is in seconds.

i. What is the acceleration of the car?

ii. What is the acceleration at t = 2?

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0.49mins

Q11bc

The velocity, in kilometres per hour, of a car as it begins to slow down is given by the equation `v(t) = 90- 30\ln(3t + 1)`

, where t is in seconds.

How long does the car take to stop?

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0.54mins

Q11d

Use the definition of the derivative to evaluate ```
\displaystyle
\lim_{h \to 0} \frac{\ln(2 + h) - \ln 2}{h}
```

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0.54mins

Q12

Consider `f(x) = \ln(\ln x)`

.

a. Determine `f'(x)`

.

b. State the domains of `f(x)`

and `f'(x)`

.

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0.00mins

Q13

Lectures
10 Videos

Introduction to Derivative of ln x with Examples

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4.47mins

Introduction to Derivative of ln x with Examples

Derivative of `y = a^x`

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4.14mins

Derivative of a^x

Derivative of `y=\ln|x|`

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1.42mins

Derivative of ln|x|

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3.57mins

Using Log Laws to Differentiate

Derivative of Logarithmic Functions with Chain and Product Rules ex1

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2.12mins

Derivative of Logarithmic Functions with Chain and Product Rules ex1

Derivative of Logarithmic Functions with Chain and Product Rules ex2

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1.41mins

Derivative of Logarithmic Functions with Chain and Product Rules ex2

Derivative of Logarithmic Functions with Chain and Product Rules ex3

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1.41mins

Derivative of Logarithmic Functions with Chain and Product Rules ex3

Derivative of Logarithmic Functions with Chain and Product Rules ex4

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1.27mins

Derivative of Logarithmic Functions with Chain and Product Rules ex4

Definition of "e"

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5.05mins

Definition of "e"