Elementary Calculus, First Midterm, Copy 2

Elementary Calculus Second Midterm, Copy 1

1. Differentiate

(1)   \begin{equation*}y =\left (3\sqrt{x} +2\right )x^{2} \end{equation*}

Solution: We know

(2)   \begin{equation*}y =\left (3\sqrt{x} +2\right )x^{2} =\left (3x^{1/2} +2\right )x^{2} \end{equation*}

We can use product rule to write

(3)   \begin{align*}\frac{dy}{dx} =\frac{d}{dx}\left (3x^{1/2} +2\right )x^{2} +\frac{d}{dx}\left (x^{2}\right )\left (3x^{1/2} +2\right ) \\ =\frac{3}{2}x^{ -1/2}\left (x^{2}\right ) +2x\left (3x^{1/2} +2\right ) \\ =\frac{15}{2}x^{3/2} +4x\end{align*}

 

We also have the option to write the original function

(4)   \begin{equation*}y =\left (3\sqrt{x} +2\right )x^{2} =\left (3x^{1/2} +2\right )x^{2} =3x^{5/2} +2x^{2} \end{equation*}

and so

(5)   \begin{equation*}\frac{dy}{dx} =\frac{15}{2}x^{3/2} +4x \end{equation*}

 

2. Differentiate

(6)   \begin{equation*}y =\frac{x^{ -1}}{x +x^{ -1}} \end{equation*}

 

Solution: We can use the quotient rule

(7)   \begin{align*}\frac{dy}{dx} =\frac{\frac{d}{dx}\left (x^{ -1}\right )\left (x +x^{ -1}\right )^{2} -\frac{d}{dx}\left (x +x^{ -1}\right )^{2}\left (x^{ -1}\right )}{\left (x +x^{ -1}\right )^{2}} \\ =\frac{ -x^{ -2}\left (x +x^{ -1}\right )^{2} -\left (1 -x^{ -2}\right )\left (x^{ -1}\right )}{\left (x +x^{ -1}\right )^{2}}\end{align*}

One can simplify (if needed).

 

3. Differentiate

(8)   \begin{equation*}y =\sqrt{\frac{x^{2} +x}{x^{2} -x}} \end{equation*}

 

Solution: We can write

(9)   \begin{equation*}y =\genfrac{(}{)}{}{}{x^{2} +x}{x^{2} -x}^{1/2} \end{equation*}

Therefore

(10)   \begin{align*}\frac{dy}{dx} =\frac{1}{2}\genfrac{(}{)}{}{}{x^{2} +x}{x^{2} -x}^{ -1/2}\frac{d}{dx}\genfrac{(}{)}{}{}{x^{2} +x}{x^{2} -x} \\ =\frac{1}{2}\genfrac{(}{)}{}{}{x^{2} +x}{x^{2} -x}^{ -1/2}\genfrac{(}{)}{}{}{\frac{d}{dx}\left (x^{2} +x\right )\left (x^{2} -x\right ) -\frac{d}{dx}\left (x^{2} -x\right )\left (x^{2} +x\right )}{\left (x^{2} -x\right )^{2}} \\ =\frac{1}{2}\genfrac{(}{)}{}{}{x^{2} +x}{x^{2} -x}^{ -1/2}\genfrac{(}{)}{}{}{\left (2x +1\right )\left (x^{2} -x\right ) -\left (2x -1\right )\left (x^{2} +x\right )}{\left (x^{2} -x\right )^{2}}\end{align*}

One can simplify when needed.

 

4. Find y^{ \prime \prime }

(11)   \begin{equation*}y =x^{1/3} \end{equation*}

Solution: We write

(12)   \begin{align*}y^{ \prime } =\frac{1}{3}x^{ -2/3} \\ y^{ \prime \prime } = -\frac{2}{9}x^{ -5/3}\end{align*}

 

5. Differentiate

(13)   \begin{equation*}y =\ln \left (e^{x} +1\right ) \end{equation*}

 

Solution: We use chain rule

(14)   \begin{align*}\frac{dy}{dx} =\frac{1}{e^{x} +1}\frac{d}{dx}\left (e^{x} +1\right ) \\ =\frac{e^{x}}{e^{x} +1}\end{align*}

 

6. Differentiate

(15)   \begin{equation*}y =e^{\sqrt{x -7}} \end{equation*}

 

Solution: We use chain rule

(16)   \begin{align*}\frac{dy}{dx} =e^{\sqrt{x -7}}\frac{d}{dx}\left (\sqrt{x -7}\right ) \\ =e^{\sqrt{x -7}}\left (\frac{1}{2}\left (x -7\right )^{ -1/2}\right )\end{align*}

 

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