3.3 The derivative

The function $f(x)$ is shown. Select all options below that describe its derivative, $\ds\diff{f}{x}$:

(a) constant (b) increasing (c) decreasing
(d) always positive (e) always negative

The function $f(x)$ is shown. Select all options below that describe its derivative, $\ds\diff{f}{x}$:

(a) constant (b) increasing (c) decreasing
(d) always positive (e) always negative

The function $f(x)$ is shown. Select all options below that describe its derivative, $\ds\diff{f}{x}$:

(a) constant (b) increasing (c) decreasing
(d) always positive (e) always negative

State, in terms of a limit, what it means for $f(x) = x^3$ to be differentiable at $x = 0$.

For which values of $x$ does $f'(x)$ not exist?

Suppose $f(x)$ is a function defined at $x=a$ with

True or false: $f'(a)=1$.

Suppose $f(x)$ is a function defined at $x=a$ with

True or false: $f'(a)=1$.

Suppose $s(t)$ is a function, with $t$ measured in seconds, and $s$ measured in metres. What are the units of $s'(t)$?

Use the definition of the derivative to find the equation of the tangent line to the curve $y(x)=x^3+5$ at the point $(1,6)$.

Use the definition of the derivative to find the derivative of $f(x)=\frac{1}{x}$.

Let $f(x) = x|x|$. Using the definition of the derivative, show that $f(x)$ is differentiable at $x = 0$.

Use the definition of the derivative to compute the derivative of the function $f(x)=\frac{2}{x+1}$.

Use the definition of the derivative to compute the derivative of the function $f(x)=\frac{1}{x^2+3}$.

Use the definition of the derivative to find the slope of the tangent line to the curve
$f(x)=x\log_{10}(2x+10)$ at the point $x=0$.

Compute the derivative of $f(x)=\frac{1}{x^2}$ directly from the definition.

Find the values of the constants $a$ and $b$ for which

is differentiable everywhere.

Remark: In the text, you have already learned the derivatives of $x^2$ and $ax+b$. In this question, you are only asked to find the values of $a$ and $b$—not to justify how you got them—so you don't have to use the definition of the derivative. However, on an exam, you might be asked to justify your answer, in which case you would show how to differentiate the two branches of $f(x)$ using the definition of a derivative.

Use the definition of the derivative to compute $f'(x)$ if $f(x) = \sqrt{1 + x}$. Where does $f'(x)$ exist?

Use the definition of the derivative to find the velocity of an object whose position is given by the function $s(t)=t^4-t^2$.

Determine whether the derivative of following function exists at $x=0$.

You must justify your answer using the definition of a derivative.

Determine whether the derivative of the following function exists at $x=0$

You must justify your answer using the definition of a derivative.

Determine whether the derivative of the following function exists at $x=0$

You must justify your answer using the definition of a derivative.

Determine whether the derivative of the following function exists at $x=1$

You must justify your answer using the definition of a derivative.

Sketch a function $f(x)$ with $f'(0)=-1$ that takes the following values:

Remark: you can't always guess the behaviour of a function from its points, even if the points seem to be making a clear pattern.

Let $p(x)=f(x)+g(x)$, for some functions $f$ and $g$ whose derivatives exist. Use limit laws and the definition of a derivative to show that $p'(x)=f'(x)+g'(x)$.

Remark: this is called the sum rule, and we'll learn more about it in Lemma 4.1.1.

Let $f(x)=2x$, $g(x)=x$, and $p(x)=f(x) \cdot g(x)$.

In Theorem 4.1.3, you'll learn a rule for calculating the derivative of a product of two functions.

There are two distinct straight lines that pass through the point $(1,-3)$ and are tangent to the curve $y = x^2$. Find equations for these two lines.

Remark: the point $(1,-3)$ does not lie on the curve $y=x^2$.

For which values of $a$ is the function

differentiable at 0?

Suppose $h(t)$ gives the height at time $t$ of the water at a dam, where the units of $t$ are hours and the units of $h$ are meters.

Suppose $p(t)$ is a function that gives the profit generated by selling $t$ widgets. What is the practical interpretation of $p'(t)$?

$T(d)$ gives the temperature of water at a particular location $d$ metres below the surface. What is the physical interpretation of $T'(d)$? Would you expect the magnitude of $T'(d)$ to be larger when $d$ is near 0, or when $d$ is very large?

$C(w)$ gives the calories in $w$ grams of a particular dish. What does $C'(w)$ describe?

The velocity of a moving object at time $t$ is given by $v(t)$. What is $v'(t)$?

The function $T(j)$ gives the temperature in degrees Celsius of a cup of water after $j$ joules of heat have been added. What is $T'(j)$?

A population of bacteria, left for a fixed amount of time at temperature $T$, grows to $P(T)$ individuals. Interpret $P'(T)$.

You hammer a small nail into a wooden wagon wheel. $R(t)$ gives the number of rotations the nail has undergone $t$ seconds after the wagon started to roll. Give an equation for how quickly the nail is rotating, measured in degrees per second.

A population of bacteria, left for a fixed amount of time at temperature $T$, grows to $P(T)$ individuals. There is one ideal temperature where the bacteria population grows largest, and the closer the sample is to that temperature, the larger the population is (unless the temperature is so extreme that it causes all the bacteria to die by freezing or boiling). How will $P'(T)$ tell you whether you are colder or hotter than the ideal temperature?