MCV4U Derivatives Practice Test (with Solutions)

Free Derivatives Practice Test

8 questions (13 parts), exactly as a free account gets them in the first Derivatives test on Martian Lab. Try each one on paper first, then open the answer.

1. Differentiate the following. Leave answers in simplified form with positive exponents only.

   1. $f(x) = \sqrt{3}x^4 + \dfrac{2}{\sqrt{x}}$
      Show answer

      $f(x) = \sqrt{3}x^4 + \dfrac{2}{\sqrt{x}} = \sqrt{3}\, x^4 + 2x^{-1/2}$ $f'(x) = 4\sqrt{3}\, x^3 - x^{-3/2}$ $= 4\sqrt{3}\, x^3 - \dfrac{1}{x^{5/2}}$
   2. $y = (7x^2 + 1)^3(2 - 5x)^7$
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      $y' = 3(7x^2{+}1)^2 \cdot 14x \cdot (2{-}5x)^7 + (7x^2{+}1)^3 \cdot 7(2{-}5x)^6(-5)$ $= (7x^2{+}1)^2(2{-}5x)^6\bigl[42x(2{-}5x) - 35(7x^2{+}1)\bigr]$ $= (7x^2{+}1)^2(2{-}5x)^6(-455x^2{+}84x{-}35)$ $= -7\,(7x^2{+}1)^2(2{-}5x)^6(65x^2{-}12x{+}5)$
2. 1. Determine the expression of $\dfrac{dy}{dx}$ for $y = \dfrac{2-x}{x^2+1}$
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      $\dfrac{dy}{dx} = \dfrac{(-1)(x^2+1) - (2-x)\cdot(2x)}{(x^2+1)^2}$ $= \dfrac{-x^2-1+2x^2-4x}{(x^2+1)^2}$ $= \dfrac{x^2-4x-1}{(x^2+1)^2}$
   2. Determine $\left.\dfrac{dy}{dx}\right|_{x=0}$
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      $\left.\dfrac{dy}{dx}\right|_{x=0} = \dfrac{-1}{1} = -1$
3. Given the graph of $y=f(x)$, give a rough sketch of the graph of $y=f'(x)$.

   

   Show solution

   Rough sketch of $y=f'(x)$.
4. If $f(x) = 2x^{10}$, what is $f^{(10)}(x)$?

   Show solution

   $f'(x) = 2 \times 10\, x^9$ $f''(x) = 2 \times 10 \times 9 \cdot x^8$ $f^3(x) = 2 \times 10 \times 9 \times 8 \cdot x^7$ $\therefore\; f^{(10)}(x) = 2 \times 10 \times 9 \times 8 \times 7 \cdots 1\, x^0$ $= 2 \times 10!$
5. Find the equation of the tangent line to $y = \bigl(\sqrt{x^2-5}\bigr)^3$ when $x = 3$

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   Using point: $x = 3,\; y = 8$ Slope: $y = (x^2-5)^{3/2}$ $y' = \dfrac{3}{2}(x^2-5)^{1/2}\cdot(2x)$ $x = 3, m = 18$ $y = mx + b$ $8 = 18 \times 3 + b,\quad b = -46$ $\therefore$ Eq. of tangent is $y = 18x - 46$
6. Find the first derivative of $x^2 + y^2 = 49xy$

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   $\dfrac{d(x^2+y^2)}{dx} = \dfrac{d(49xy)}{dx}, \qquad \dfrac{dx^2}{dx} + \dfrac{dy^2}{dx} = 49\,\dfrac{d(xy)}{dx}$ $2x + \dfrac{dy^2}{dy}\cdot\dfrac{dy}{dx} = 49\!\left(\dfrac{dx}{dx}\cdot y + x\cdot\dfrac{dy}{dx}\right)$ $2x + 2yy' = 49y + 49xy'$ $(2y - 49x)\,y' = 49y - 2x$ $y' = \dfrac{49y - 2x}{2y - 49x}$
7. The population of stray cats in Windsor can be described by $P(x) = -x^2 + 18x + 19$, where $x$ is the number of years since the year 2010 (which is the first year this data is being tracked).

   1. What is the meaning of $P'(x)$ for this scenario? Be sure to include the units in your explanation.
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      The rate of change of stray cats population with respect to time in no. of cats per year.
   2. If $P'(11) = -4$, explain what this means.
      Show answer

      In year 2021 (2010+11), the stray cats population decreases by 4 cats per year.
   3. If this year, we decided to change the model to $P(x) = x^2 + 2x + 19$, what does that mean?
      Show answer

      The rate of change of the population would change.
8. 1. Given $f(x) = \dfrac{x^2+7}{x-3}$, find the value(s) of $x$ such that $f'(x) = 0$
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      $f'(x) = \dfrac{2x(x-3)-(x^2+7)}{(x-3)^2} = \dfrac{x^2-6x-7}{(x-3)^2} = 0$ $(x-7)(x+1) = 0$ $x = 7 \text{ or } -1$
   2. Find all value(s) of $x$ such that $f'(x)$ is negative. Justify your solution over a Real Number line or a table.
      Show solution

      $\dfrac{x^2-6x-7}{(x-3)^2} < 0, \quad x^2 - 6x - 7 < 0$ $(x-7)(x+1) < 0$ $-1 < x < 7, \quad x \in \mathbb{R}$

This Unit on Martian Lab

The lessons go from what a derivative is through every differentiation rule to applications, including business and economics, and implicit differentiation.

1. Derivatives
2. Sketching Derivative Functions
3. Derivative Rules
4. Product and Quotient Rules
5. Chain Rule
6. Applications of Derivatives
7. Higher Derivatives
8. Applications in Business and Economics
9. Implicit Differentiation

Then 4 full Derivatives practice tests to check yourself.

Derivatives FAQ

Derivatives Topics

One idea at a time: a short explanation, then five worked examples.

• Chain Rule

Start Practicing

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