MCV4U · Unit 2 test review

Derivative Rules: Test Review

Everything for the Unit 2 test on one page: the key facts from all 6 lessons, the mistakes that cost the most marks, and a practice test that points you back to the lessons you need.

Key facts

Power, Constant, Sum and Difference Rules

Power rule: $\frac{d}{d x} (x^{n}) = n x^{n - 1}$ for natural and rational exponents, including negative ones.
Constant rule: $\frac{d}{d x} (c) = 0$ . Constant multiple rule: $\frac{d}{d x} [k f (x)] = k f^{'} (x)$ .
Sum and difference rules: $(f \pm g)^{'} (x) = f^{'} (x) \pm g^{'} (x)$ , so polynomials are differentiated term by term.
Rewrite before differentiating: $x = x^{\frac{1}{2}}$ and $\frac{1}{x ^{n}} = x^{- n}$ .
Tangent slope at $x = a$ is $f^{'} (a)$ ; the tangent line is $y - f (a) = f^{'} (a) (x - a)$ .
To find where the slope equals $m$ , solve $f^{'} (x) = m$ , then use $f (x)$ to find each point's $y$ -coordinate.

The Product Rule

Product rule: $(f g)^{'} = f^{'} g + f g^{'}$ .
$(f g)^{'} \neq = f^{'} g^{'}$ : check with $f = g = x$ , where $(x^{2})^{'} = 2 x$ but $1 \cdot 1 = 1$ .
Leibniz form: $\frac{d}{d x} (uv) = \frac{d u}{d x} v + u \frac{d v}{d x}$ .
For three factors: $(f g h)^{'} = f^{'} g h + f g^{'} h + f g h^{'}$ .
To find $f^{'} (a)$ , substitute $a$ into each of $f, g, f^{'}, g^{'}$ ; there is no need to simplify first.

The Chain Rule

Chain rule (Leibniz): $\frac{d y}{d x} = \frac{d y}{d u} \cdot \frac{d u}{d x}$ .
Chain rule (function notation): $[f (g (x))]^{'} = f^{'} (g (x)) \cdot g^{'} (x) .$
Power of a function: $\frac{d}{d x} [g (x)]^{n} = n [g (x)]^{n - 1} g^{'} (x) .$
Radical: $\frac{d}{d x} g (x) = \frac{g ^{'} ( x )}{2 g ( x )}$ .
Linear inside: $\frac{d}{d x} (a x + b)^{n} = an (a x + b)^{n - 1}$ ; the $a$ is easy to forget.

Derivatives of Rational Functions

Quotient rule: $(\frac{f}{g})^{'} = \frac{f ^{'} g - f g ^{'}}{g ^{2}}$ , where $g \neq = 0$ .
Alternative: $\frac{f}{g} = f \cdot g^{- 1}$ , then use the product and chain rules.
Constant numerator: $\frac{d}{d x} [\frac{k}{g ( x )}] = - \frac{k g ^{'} ( x )}{[ g ( x ) ] ^{2}} .$
$(\frac{f}{g})^{'} \neq = \frac{f ^{'}}{g ^{'}}$ , and the numerator order $f^{'} g - f g^{'}$ matters.
A rational function's derivative does not exist where its denominator is $0$ ; horizontal tangents occur where the numerator of $f^{'}$ is $0$ .

Derivatives of Sine and Cosine

$\frac{d}{d x} (sin x) = cos x$ and $\frac{d}{d x} (cos x) = - sin x$ , with $x$ in radians.
Chain rule versions: $\frac{d}{d x} sin (g (x)) = cos (g (x)) g^{'} (x)$ and $\frac{d}{d x} cos (g (x)) = - sin (g (x)) g^{'} (x) .$
$\frac{d}{d x} sin (k x) = k cos (k x)$ and $\frac{d}{d x} cos (k x) = - k sin (k x)$ .
$\frac{d}{d x} tan x = \frac{1}{cos ^{2} x} = sec^{2} x$ (quotient rule plus $sin^{2} x + cos^{2} x = 1$ ).
Repeated derivatives of $sin x$ cycle: $cos x, - sin x, - cos x, sin x$ .

Derivatives of Exponential Functions

$\frac{d}{d x} (e^{x}) = e^{x}$ : for $y = e^{x}$ , slope equals height at every point.
$\frac{d}{d x} (a^{x}) = a^{x} ln a$ for $a > 0$ , $a \neq = 1$ ; so $\frac{f ^{'} ( x )}{f ( x )} = ln a$ is constant.
Chain rule with $e$ : $\frac{d}{d x} e^{g (x)} = e^{g (x)} g^{'} (x)$ , for example $\frac{d}{d x} e^{k x} = k e^{k x}$ .
$ln x = lo g_{e} x$ is the inverse of $e^{x}$ : $e^{l n x} = x$ (for $x > 0$ ) and $ln (e^{x}) = x$ .
The graphs of $y = e^{x}$ and $y = ln x$ are reflections in $y = x$ ; $(0, 1) \leftrightarrow (1, 0)$ and $(1, e) \leftrightarrow (e, 1)$ .
$e \approx 2.71828$ is the base between $2$ and $3$ whose tangent slope at $x = 0$ is exactly $1$ .

Where marks get lost

Differentiating $\frac{1}{x ^{2}}$ as $\frac{1}{2 x}$ . Rewrite it as $x^{- 2}$ first; its derivative is $- 2 x^{- 3} = - \frac{2}{x ^{3}}$ . (Power, Constant, Sum and Difference Rules)
Subtracting one from a negative exponent in the wrong direction: $x^{- 2}$ becomes $- 2 x^{- 3}$ , not $- 2 x^{- 1}$ . One less than $- 2$ is $- 3$ . (Power, Constant, Sum and Difference Rules)
Keeping the constant term: the derivative of $x^{2} + 7$ is $2 x$ , not $2 x + 7$ . (Power, Constant, Sum and Difference Rules)
Differentiating a product factor by factor, for example claiming $\frac{d}{d x} [x^{2} (x^{3} + 1)] = 2 x \cdot 3 x^{2}$ . Either expand first or use the product rule from the next lesson. (Power, Constant, Sum and Difference Rules)
Substituting into $f^{'} (x)$ instead of $f (x)$ to find the $y$ -coordinate of a point of tangency. $f^{'} (a)$ is the slope; $f (a)$ is the height. (Power, Constant, Sum and Difference Rules)
Writing $(f g)^{'} = f^{'} g^{'}$ . Test it on $x \cdot x$ : it predicts a derivative of $1$ , but $(x^{2})^{'} = 2 x$ . (The Product Rule)
Writing only one of the two terms, for example $f^{'} g$ , and forgetting $f g^{'}$ . Each factor needs its turn to be differentiated. (The Product Rule)
Dropping brackets: $3 (2 x - 1)$ written as $3 \cdot 2 x - 1$ . Bracket every binomial before multiplying. (The Product Rule)
Leaving a radical as $x$ and then differentiating it as if it were $x$ . Rewrite $x$ as $x^{\frac{1}{2}}$ before applying the power rule to that factor. (The Product Rule)
Forgetting the derivative of the inside: writing $\frac{d}{d x} (2 x - 3)^{4} = 4 (2 x - 3)^{3}$ and losing the factor of $2$ . (The Chain Rule)
Differentiating the inside within the outer factor: writing $5 (6 x)^{4}$ for the derivative of $(3 x^{2} + 1)^{5}$ instead of $5 (3 x^{2} + 1)^{4} \cdot 6 x$ . (The Chain Rule)
Getting the exponent wrong for a root: the derivative of $(x + 5)^{\frac{1}{2}}$ is $\frac{1}{2} (x + 5)^{- \frac{1}{2}}$ , not $\frac{1}{2} (x + 5)^{\frac{1}{2}}$ . (The Chain Rule)
In Leibniz form, evaluating $\frac{d y}{d u}$ at the $x$ -value instead of at the matching $u$ -value. Find $u$ first, then substitute it into $\frac{d y}{d u}$ . (The Chain Rule)
Reversing the numerator to $f g^{'} - f^{'} g$ . That gives exactly the negative of the correct derivative. Always start with the derivative of the top. (Derivatives of Rational Functions)
Forgetting to square the denominator, or squaring the numerator as well. (Derivatives of Rational Functions)
Writing $(\frac{f}{g})^{'} = \frac{f ^{'}}{g ^{'}}$ . Test it on $\frac{x ^{2}}{x} = x$ : it would give $\frac{2 x}{1} = 2 x$ , but the derivative of $x$ is $1$ . (Derivatives of Rational Functions)
Cancelling terms instead of factors, for example crossing out an $x^{2}$ that appears in a sum on top and in $(x^{2} + 1)^{2}$ on the bottom. (Derivatives of Rational Functions)
Reporting a derivative at a value that makes the denominator $0$ . The function itself is undefined there, so the derivative does not exist. (Derivatives of Rational Functions)
Getting the sign wrong for cosine: $\frac{d}{d x} cos x$ is $- sin x$ , not $sin x$ . At $x = \frac{π}{2}$ the cosine graph is falling, so its slope must be negative there. (Derivatives of Sine and Cosine)
Evaluating with the calculator in degree mode. $cos 1$ in degrees is about $0.9998$ ; in radians it is about $0.5403$ . (Derivatives of Sine and Cosine)
Forgetting the chain factor: $\frac{d}{d x} sin (3 x)$ is $3 cos (3 x)$ , not $cos (3 x)$ . (Derivatives of Sine and Cosine)
Writing $\frac{d}{d x} sin (2 x) = 2 cos x$ or $cos 2$ . The inside stays exactly as it was: $2 cos (2 x)$ . (Derivatives of Sine and Cosine)
Treating $sin^{2} x$ as $sin (x^{2})$ , or the reverse. Decide which operation happens last before choosing the outer function. (Derivatives of Sine and Cosine)
Using the power rule on an exponential: writing $\frac{d}{d x} e^{x} = x e^{x - 1}$ or $\frac{d}{d x} 2^{x} = x \cdot 2^{x - 1}$ . The variable is in the exponent, so the exponential rules apply. (Derivatives of Exponential Functions)
Forgetting the factor $ln a$ : $\frac{d}{d x} 2^{x}$ is $2^{x} ln 2$ , not $2^{x}$ . Only base $e$ has no extra factor, because $ln e = 1$ . (Derivatives of Exponential Functions)
Forgetting the chain factor: $\frac{d}{d x} e^{3 x}$ is $3 e^{3 x}$ , not $e^{3 x}$ . (Derivatives of Exponential Functions)
Differentiating the exponent instead of multiplying by its derivative: $\frac{d}{d x} e^{x^{2}}$ is $2 x e^{x^{2}}$ , not $e^{2 x}$ . The exponential is copied exactly. (Derivatives of Exponential Functions)
Pressing the LOG key (base $10$ ) instead of LN (base $e$ ) on a calculator when evaluating $ln a$ . (Derivatives of Exponential Functions)

Practice test

24 questions from across the unit. Each one shows the lesson it comes from.

Power, Constant, Sum and Difference Rules
1
What is $\frac{d}{d x} (x^{7})$ ?
Power, Constant, Sum and Difference Rules
2
For $y = \frac{10}{x ^{2}}$ , find $\frac{d y}{d x}$ at $x = 2$ . Give the exact decimal value.
Power, Constant, Sum and Difference Rules
3
The tangent line to $y = x^{2} + 3 x$ at $x = 1$ has equation $y = m x + b$ . Find $b$ .
Power, Constant, Sum and Difference Rules
4
Which rule justifies differentiating $x^{2} + x^{5}$ as $2 x + 5 x^{4}$ , one term at a time? (Two words.)
The Product Rule
5
Which is $\frac{d}{d x} [x^{3} (2 x + 1)]$ ?
The Product Rule
6
A ticket price is $p (t) = 25 + t$ dollars and the number of tickets sold per week is $n (t) = 800 - 20 t$ , with $t$ in weeks. Revenue is $R = p n$ . Find $R^{'} (5)$ in dollars per week.
dollars/week
The Product Rule
7
Put the steps in order to find $h^{'} (a)$ for $h (x) = f (x) g (x)$ using the product rule.
1. 1Differentiate each factor to get $f^{'}$ and $g^{'}$
2. 2Substitute $x = a$ and evaluate
3. 3Write $h^{'} (x) = f^{'} (x) g (x) + f (x) g^{'} (x)$
4. 4Identify the two factors $f$ and $g$
The Product Rule
8
Name the rule that says $\frac{d}{d x} (uv) = \frac{d u}{d x} v + u \frac{d v}{d x}$ .
The Chain Rule
9
For $f (x) = (2 x - 3)^{4}$ , find $f^{'} (2)$ .
The Chain Rule
10
Let $y = u^{3}$ and $u = x^{2} - 4 x$ . Find $\frac{d y}{d x}$ at $x = 1$ .
The Chain Rule
11
Put the steps of the chain rule in Leibniz form in order.
1. 1Replace $u$ with $g (x)$ so the answer is in terms of $x$
2. 2Multiply: $\frac{d y}{d x} = \frac{d y}{d u} \cdot \frac{d u}{d x}$
3. 3Find $\frac{d y}{d u}$ and $\frac{d u}{d x}$
4. 4Identify the inner function $u = g (x)$
5. 5Write the outer function as $y = f (u)$
The Chain Rule
12
Name the rule that says $\frac{d y}{d x} = \frac{d y}{d u} \cdot \frac{d u}{d x}$ .
Derivatives of Rational Functions
13
For $f (x) = \frac{3 x}{x + 2}$ , find $f^{'} (1)$ . Give your answer to 3 decimal places.
Derivatives of Rational Functions
14
Match each quotient to the way you would rewrite it as a product (or a single power).
1. A. $5 (2 x + 1)^{- 1}$
2. B. $x (x + 4)^{- 1}$
3. C. $x^{- \frac{1}{2}}$
4. D. $x^{- 3}$
$\frac{1}{x ^{3}}$
$\frac{5}{2 x + 1}$
$\frac{x}{x + 4}$
$\frac{1}{x}$
Derivatives of Rational Functions
15
For $f (x) = \frac{x + 1}{x - 5}$ , where does $f^{'} (x)$ fail to exist?
Derivatives of Rational Functions
16
Name the rule $(\frac{f}{g})^{'} = \frac{f ^{'} g - f g ^{'}}{g ^{2}}$ .
Derivatives of Sine and Cosine
17
What is $\frac{d}{d x} (cos x)$ ?
Derivatives of Sine and Cosine
18
Match each function to its derivative.
1. A. $- sin x$
2. B. $cos x$
3. C. $sin x$
4. D. $2 cos (2 x)$
$sin x$
$cos x$
$sin (2 x)$
$- cos x$
Derivatives of Sine and Cosine
19
Why must $x$ be in radians for $\frac{d}{d x} sin x = cos x$ to hold?
Derivatives of Sine and Cosine
20
In what unit must angles be measured for the rule $\frac{d}{d x} sin x = cos x$ to be true?
Derivatives of Exponential Functions
21
What is $\frac{d}{d x} (e^{x})$ ?
Derivatives of Exponential Functions
22
Match each function to its derivative.
1. A. $1 0^{x} ln 10$
2. B. $2 x e^{x^{2}}$
3. C. $4 e^{4 x}$
4. D. $e^{x}$
$e^{x}$
$e^{4 x}$
$1 0^{x}$
$e^{x^{2}}$
Derivatives of Exponential Functions
23
For $f (x) = e^{x^{2}}$ , find $f^{'} (1)$ . Give your answer to 3 decimal places.
Derivatives of Exponential Functions
24
For $f (x) = a^{x}$ ( $a > 0$ , $a \neq = 1$ ), what can you say about the ratio $\frac{f ^{'} ( x )}{f ( x )}$ ?

Endless drill

Fresh numbers every time, for the calculation questions in this unit.

Endless drillnew numbers every time

Generating a problem…