Defining the Derivative and Using Derivative Notation Practice

Sample questions

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1. Sample 1difficulty 1/5

   Which is NOT standard notation for the derivative of $y$ with respect to $x$?

   • A

     $y \div x$

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   • B

     $y'$

   • C

     $D_x y$

   • D

     $\dfrac{dy}{dx}$

   Why

   Division by $x$ is not the derivative.

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2. Sample 2difficulty 2/5

   The derivative $f'(a)$ is defined as

   • A

     $f(a) \cdot a$

   • B

     $\dfrac{f(b)-f(a)}{b-a}$

   • C

     $\displaystyle\lim_{h\to 0}\dfrac{f(a+h)-f(a)}{h}$

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   • D

     $\dfrac{f(a+h)-f(a)}{h}$

   Why

   The derivative is the limit of the difference quotient.

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3. Sample 3difficulty 3/5

   $\displaystyle\lim_{h\to 0}\dfrac{\sin(\pi/2 + h) - 1}{h}$ equals

   • A

     Does not exist

   • B

     $1$

   • C

     $-1$

   • D

     $0$

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   Why

   This is $f'(\pi/2)$ for $f(x) = \sin x$. $f'(x) = \cos x$; $\cos(\pi/2) = 0$.

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4. Sample 4difficulty 3/5

   The derivative at $x=a$ is the limit of secant slopes, defined as:

   • A

     $\lim_{h \to 0} \dfrac{f(a+h)-f(a)}{h}$

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   • B

     $\lim_{x \to a} f(x)$

   • C

     $f(a)$

   • D

     $\dfrac{f(a+h)-f(a)}{h}$

   Why

   Definition of derivative: $f'(a) = \lim_{h\to 0} \frac{f(a+h) - f(a)}{h}$.

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5. Sample 5difficulty 3/5

   Use the definition to find $f'(2)$ for $f(x) = x^2 + 1$.

   • A

     $2$

   • B

     $5$

   • C

     $3$

   • D

     $4$

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   Why

   $\lim_{h\to 0}((2+h)^2+1 - 5)/h = \lim (4h+h^2)/h = 4$.

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