Section 1.5 Practice Gateway Exam 5
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.
\(y = 7x^{6} + 2x^{3.1}-4x+ \dfrac{5}{x^6}\)
Solution.
\begin{align*}
\frac{dy}{dx} \amp = \frac{d}{dx}[7x^6 + 2x^{3.1} - 4x + 5x^{-6}]\\
\amp = 42x^5 + 6.2x^{2.1} - 4 - 30x^{-7}
\end{align*}
2.
\(g(x) = x^4 \ln(6x^5)+ 10e^3\)
Solution.
\begin{align*}
g'(x) \amp = x^4 \cdot \frac{d}{dx}[\ln(6x^5)] + \ln(6x^5) \cdot \frac{d}{dx}[x^4]\\
\amp = x^4 \cdot \frac{1}{6x^5} \cdot \frac{d}{dx}[6x^5] + \ln(6x^5)\cdot 4x^3\\
\amp = x^4 \cdot \frac{1}{6x^5} \cdot 30x^4 + \ln(6x^5)\cdot 4x^3
\end{align*}
3.
\(r(\theta) = 8(\sin(4\theta))^{-10}\)
Solution.
\begin{align*}
r'(\theta) \amp = -80 (\sin(4\theta))^{-11}\cdot \frac{d}{d\theta}[\sin(4\theta)]\\
\amp = -80 (\sin(4\theta))^{-11}\cdot \cos(4\theta) \cdot \frac{d}{d\theta}[4\theta]\\
\amp = -80 (\sin(4\theta))^{-11}\cdot \cos(4\theta) \cdot 4
\end{align*}
4.
\(f(x) = \ln(6) 7^{100x}\)
Solution.
\begin{align*}
f'(x) \amp = \ln(6) \cdot \ln(7)\cdot 7^{100x}\cdot \frac{d}{dx}[100x]\\
\amp = \ln(6)\cdot \ln(7)\cdot 7^{100x}\cdot 100
\end{align*}
5.
\(f(w) = \dfrac{w^6}{\arcsin(w)}\)
Solution.
\begin{align*}
f'(w) \amp = \frac{\arcsin(w)\cdot \frac{d}{dw}[w^6] - w^6 \cdot \frac{d}{dw}[\arcsin(w)]}{(\arcsin(w))^2}\\
\amp = \cfrac{\arcsin(w)\cdot (6w^5) - w^6 \cdot \frac{1}{\sqrt{1-w^2}}}{(\arcsin(w))^2}
\end{align*}
6.
\(g(t) = \arctan(7t^4)\)
Solution.
\begin{align*}
g'(t) \amp = \frac{1}{1+(7t^4)^2} \cdot \frac{d}{dt}[7t^4]\\
\amp = \frac{1}{1+(7t^4)^2} \cdot 28t^3
\end{align*}
7.
\(h(z) = \cos(60z^2 + 100z)\)
Solution.
\begin{align*}
h'(z) \amp = -\sin(60z^2 + 100z) \cdot \frac{d}{dz}[60z^2+100z] \\
\amp = -\sin(60z^2 + 100z) \cdot (120z+100)
\end{align*}
8.
\(h(x) = 6e^{5x} + \sqrt{x^5-6x} \)
Solution.
\begin{align*}
h'(x) \amp = \frac{d}{dx}\left[6e^{5x} + (x^5-6x)^{1/2}\right]\\
\amp = 6e^{5x} \cdot \frac{d}{dx}[5x] + \frac{1}{2}(x^5-6x)^{-1/2} \cdot \frac{d}{dx}[x^5-6x]\\
\amp = 6e^{5x} \cdot 5 + \frac{1}{2}(x^5-6x)^{-1/2}\cdot (5x^4-6)
\end{align*}