Section 1.6 Practice Gateway Exam 6
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.
\(f(x) = 3x^{4} - x^{3.6} + \sqrt{12}\)
Solution.
\begin{align*}
f'(x) \amp = 12x^3 - 3.6x^{2.6}\\
\end{align*}
2.
\(g(x) = \cos(2x) + 2 \sin(x)\)
Solution.
\begin{align*}
g'(x) \amp = -\sin(2x) \cdot \frac{d}{dx}[2x] + 2\cos(x)\\
\amp = -\sin(2x) \cdot 2 + 2\cos(x)
\end{align*}
3.
\(h(z) = 2e^z z^3\)
Solution.
\begin{align*}
h'(z) \amp = 2e^z \cdot \frac{d}{dz}[z^3] + z^3 \cdot \frac{d}{dz}[2e^z]\\
\amp = 2e^z \cdot 3z^2 + z^3\cdot 2e^z
\end{align*}
4.
\(y = \sqrt{x^3+e^2}\)
Solution.
\begin{align*}
\frac{dy}{dx} \amp = \frac{d}{dx}[(x^3+e^2)^{1/2}]\\
\amp = \frac{1}{2} (x^3+e^2)^{-1/2}\cdot \frac{d}{dx}[x^3+e^2]\\
\amp = \frac{1}{2} (x^3+e^2)^{-1/2} \cdot 3x^2
\end{align*}
5.
\(s(t) = \ln(t^2+2.8)\)
Solution.
\begin{align*}
s'(t) \amp = \frac{1}{t^2+2.8}\cdot \frac{d}{dt}[t^2+2.8]\\
\amp = \frac{1}{t^2+2.8} \cdot 2t
\end{align*}
6.
\(m(t) = \dfrac{t-2}{e^t}\)
Solution.
\begin{align*}
m'(t) \amp = \frac{e^t \cdot \frac{d}{dt}[t-2] - (t-2)\cdot \frac{d}{dt}[e^t]}{(e^t)^2}\\
\amp = \frac{e^t - (t-2)\cdot e^t}{e^{2t}}
\end{align*}
7.
\(r(x) = e^{2.7x} + \dfrac{1}{x}\)
Solution.
\begin{align*}
r'(x) \amp = \frac{d}{dx}[e^{2.7x} + x^{-1}]\\
\amp = e^{2.7x}\cdot \frac{d}{dx}[2.7x]-x^{-2}\\
\amp = e^{2.7x}\cdot 2.7 -x^{-2}
\end{align*}
8.
\(y(x) = 2(6^x) - e^{\ln(x)}\)
Solution.
\begin{align*}
y'(x) \amp = \frac{d}{dx}[2(6^x) - x]\\
\amp = 2\cdot \ln(6) \cdot 6^x - 1
\end{align*}