Section 1.7 Practice Gateway Exam 7
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.
\(f(x) = x^{6} - 2x^{7.1} + 5x + 3\)
Solution.
\begin{align*}
f'(x) \amp = 6x^5 - 2\cdot 7.1x^{6.1} + 5
\end{align*}
2.
\(g(t) = 6\ln(t^8 + 5t)\)
Solution.
\begin{align*}
g'(t) \amp = 6\cdot \frac{1}{t^8 + 5t}\cdot \frac{d}{dt}[t^8+5t]\\
\amp = \frac{6}{t^8+5t} \cdot (8t^7 + 5)
\end{align*}
3.
\(r(\theta) = \cos(2\theta^5+\theta)\)
Solution.
\begin{align*}
r'(\theta) \amp = -\sin(2\theta^5 +\theta) \cdot \frac{d}{d\theta}[2\theta^5 + \theta]\\
\amp = -\sin(2\theta^5 + \theta) \cdot (10\theta^4 + 1)
\end{align*}
4.
\(f(z) = z \arctan(7z)\)
Solution.
\begin{align*}
f'(z) \amp = z \cdot \frac{d}{dz}[\arctan(7z)] + \arctan(7z)\cdot \frac{d}{dz}[z]\\
\amp = z\cdot \frac{1}{1+(7z)^2}\cdot \frac{d}{dz}[7z] + \arctan(7z)\cdot 1\\
\amp = \frac{z}{1+49z^2}\cdot 7 + \arctan(7z)
\end{align*}
5.
\(f(w) = \dfrac{4w^3-6w}{\sin(w)}\)
Solution.
\begin{align*}
f'(w) \amp = \frac{\sin(w) \cdot \frac{d}{dw}[4w^3-6w] - (4w^3-6w)\cdot \frac{d}{dw}[\sin(w)]}{(\sin(w))^2}\\
\amp = \frac{\sin(w)\cdot(12w^2-6) - (4w^3-6w)\cdot \cos(w)}{\sin^2(w)}
\end{align*}
6.
\(g(x) = x^6 e^{3x}\)
Solution.
\begin{align*}
g'(x) \amp = x^6 \cdot \frac{d}{dx}[e^{3x}] + e^{3x}\cdot\frac{d}{dx}[x^6]\\
\amp = x^6\cdot e^{3x}\cdot \frac{d}{dx}[3x] + e^{3x}\cdot 6x^5\\
\amp = x^6 \cdot e^{3x}\cdot 3 + e^{3x}\cdot 6x^5
\end{align*}
7.
\(y = \ln(9) 4^{2x+5}\)
Solution.
\begin{align*}
\frac{dy}{dx} \amp = \ln(9) \cdot \ln(4)\cdot 4^{2x+5}\cdot \frac{d}{dx}[2x+5]\\
\amp = \ln(9)\cdot \ln(4) \cdot 4^{2x+5}\cdot 2
\end{align*}
8.
\(h(x) = \dfrac{1}{x^3} + \sqrt[3]{x} + 3x - \dfrac{1}{3} \pi\)
Solution.
\begin{align*}
h'(x) \amp = \frac{d}{dx}\left[x^{-3} + x^{1/3} +3x - \frac{1}{3}\pi \right]\\
\amp = -3x^{-4} + \frac{1}{3}x^{-2/3} + 3
\end{align*}