Section 1.2 Practice Gateway Exam 2
The use of a calculator or any computer algebra system is not permitted on this test. No partial credit is available. A score of at least 7 correct is required to pass this gateway.
Find the derivative of each of the following functions:
Exercises Exercises
1.
\(f(x) = x^{16} + 2x + \dfrac{1}{x^4}-8e^2\)
Solution.
\begin{align*}
f'(x) \amp = \frac{d}{dx}\left[ x^{16} +2x +x^{-4} -8e^2\right]\\
\amp = 16x^{15} + 2 -4x^{-5}
\end{align*}
2.
\(y = (3x+\sin(x))e^{2x}\)
Solution.
\begin{align*}
\frac{dy}{dx} \amp = (3x+\sin(x))\frac{d}{dx}[e^{2x}] + e^{2x} \frac{d}{dx}[3x+\sin(x)]\\
\amp = (3x+\sin(x))\cdot e^{2x} \cdot 2 + e^{2x}(3+\cos(x))
\end{align*}
3.
\(r(x) = \ln(\cos(x)) + \ln(e^x)\)
Solution.
\begin{align*}
r'(x) \amp = \frac{d}{dx}[\ln(\cos(x)) + x]\\
\amp = \frac{1}{\cos(x)}\cdot \frac{d}{dx}[\cos(x)]+1\\
\amp = \frac{1}{\cos(x)} \cdot(-\sin(x))+1
\end{align*}
4.
\(f(t) = \arctan(t^2)\)
Solution.
\begin{align*}
f'(t) \amp = \frac{1}{1+(t^2)^2} \cdot \frac{d}{dt}[t^2]\\
\amp = \frac{2t}{1+t^4}
\end{align*}
5.
\(f(t) = \dfrac{t^2+2}{\sqrt{t}}\)
Solution.
\begin{align*}
f'(t) \amp = \frac{d}{dt} \left[\frac{t^2}{\sqrt{t}} + \frac{2}{\sqrt{t}} \right]\\
\amp = \frac{d}{dt} \left[ t^{3/2} + 2t^{-1/2}\right]\\
\amp = \frac{3}{2}t^{1/2} + 2 \cdot \frac{-1}{2}t^{-3/2}\\
\amp = \frac{3}{2}t^{1/2} -t^{-3/2}
\end{align*}
6.
\(g(x) = (e^x+12x)^{12}\)
Solution.
\begin{align*}
g'(x) \amp = 12(e^x + 12x)^{11}\cdot \frac{d}{dx} [e^x + 12x]\\
\amp = 12(e^x + 12x)^{11} \cdot(e^x+12)
\end{align*}
7.
\(h(t) = \cos^2(3t+5)\)
Solution.
\begin{align*}
h'(t) \amp = \frac{d}{dt}[ (\cos(3t+5))^2]\\
\amp = 2\cos(3t+5) \cdot \frac{d}{dt}[ \cos(3t+5)]\\
\amp = 2\cos(3t+5) \cdot (-\sin(3t+5))\cdot 3
\end{align*}
8.
\(h(x) = 3^x \tan(x)\)
Solution.
\begin{align*}
h'(x) \amp = 3^x \cdot \frac{d}{dx}[\tan(x)] + \tan(x) \cdot \frac{d}{dx}[3^x]\\
\amp = 3^x \cdot \sec^2(x) + \tan(x) \cdot \ln(3) \cdot 3^x
\end{align*}