New Zealand
Level 8 - NCEA Level 3

# Basic derivative of Logs

## Interactive practice questions

Consider the graph of $y=\ln x$y=lnx.

a

Is the function increasing or decreasing?

Increasing

A

Decreasing

B

Increasing

A

Decreasing

B
b

Is the gradient to the curve negative at any point on the curve?

No

A

Yes

B

No

A

Yes

B
c

Which of the following best completes this sentence?

"As $x$x increases, the gradient of the tangent..."

decreases at a constant rate.

A

increases at an increasing rate.

B

increases at a constant rate.

C

decreases at an increasing rate.

D

increases at a decreasing rate.

E

decreases at a decreasing rate.

F

decreases at a constant rate.

A

increases at an increasing rate.

B

increases at a constant rate.

C

decreases at an increasing rate.

D

increases at a decreasing rate.

E

decreases at a decreasing rate.

F
d

Which of the following best completes the sentence?

"As $x$x gets closer and closer to $0$0, the gradient of the tangent..."

increases towards a fixed value.

A

decreases towards $-\infty$.

B

decreases towards $0$0.

C

increases towards $\infty$.

D

increases towards a fixed value.

A

decreases towards $-\infty$.

B

decreases towards $0$0.

C

increases towards $\infty$.

D
e

We have found that the gradient function must be a strictly positive function, and it must also be a function that decreases at a decreasing rate. What kind of function could it be?

Quadratic, of the form $y'=ax^2$y=ax2.

A

Exponential, of the form $y'=a^{-x}$y=ax.

B

Linear, of the form $y=ax$y=ax.

C

Hyperbolic, of the form $y'=\frac{a}{x}$y=ax.

D

Quadratic, of the form $y'=ax^2$y=ax2.

A

Exponential, of the form $y'=a^{-x}$y=ax.

B

Linear, of the form $y=ax$y=ax.

C

Hyperbolic, of the form $y'=\frac{a}{x}$y=ax.

D
Easy
Approx 3 minutes

In this problem we will find the derivative of $y=\ln x$y=lnx using the chain rule.

Consider the function $f\left(x\right)=\ln x$f(x)=lnx.

### Outcomes

#### M8-11

Choose and apply a variety of differentiation, integration, and antidifferentiation techniques to functions and relations, using both analytical and numerical methods

#### 91578

Apply differentiation methods in solving problems