Section 1.8 Practice Gateway Exam 8
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.
\(f(x) = 3x^4 - 2x^3 + 8e^5\)
Solution.
\begin{align*}
f'(x) \amp = 12x^3 - 6x^2
\end{align*}
2.
\(g(\theta) = \sin(2\theta) + 3 \cos(\theta)\)
Solution.
\begin{align*}
g'(\theta) \amp = \cos(2\theta) \cdot \frac{d}{d\theta}[2\theta] -3\sin(\theta)\\
\amp = \cos(2\theta) \cdot 2 - 3\sin(\theta)
\end{align*}
3.
\(h(x) = (x^2+1) \ln(5x)\)
Solution.
\begin{align*}
h'(x) \amp = (x^2+1)\cdot \frac{d}{dx}[\ln(5x)] + \ln(5x) \cdot \frac{d}{dx}[x^2+1]\\
\amp = (x^2+1)\cdot \frac{1}{5x}\cdot \frac{d}{dx}[5x] + \ln(5x)\cdot (2x)\\
\amp = (x^2+1) \cdot \frac{1}{5x} \cdot 5 + \ln(5x)\cdot 2x
\end{align*}
4.
\(r(x) = \dfrac{1}{\sqrt{1+x^2}}\)
Solution.
\begin{align*}
f'(x) = \frac{d}{dx}[(1+x^2)^{-1/2}]\\
\amp = -\frac{1}{2}(1+x^2)^{-3/2} \cdot \frac{d}{dx}[1+x^2]\\
\amp = -\frac{1}{2}(1+x^2)^{-3/2} \cdot 2x
\end{align*}
5.
\(s(t) = \ln(t^3+t)\)
Solution.
\begin{align*}
s'(t) \amp = \frac{1}{t^3+t} \cdot \frac{d}{dt}[t^3+t]\\
\amp = \frac{1}{t^3+t} \cdot (3t^2+1)
\end{align*}
6.
\(y = \dfrac{x^3+2x^2}{x^2+1}\)
Solution.
\begin{align*}
\frac{dy}{dx} \amp = \frac{(x^2+1)\cdot \frac{d}{dx}[x^3+2x^2] - (x^3+2x)\cdot \frac{d}{dx}[x^2+1]}{(x^2+1)^2}\\
\amp = \frac{(x^2+1)\cdot(3x^2+4x) - (x^3+2x^2)\cdot 2x}{(x^2+1)^2}
\end{align*}
7.
\(r(x) = e^{3x+2} + 4\tan(5x)\)
Solution.
\begin{align*}
r'(x) \amp = e^{3x+2} \cdot \frac{d}{dx}[3x+2] + 4\sec^2(5x)\cdot \frac{d}{dx}[5x]\\
\amp = e^{3x+2} \cdot 3 + 4\sec^2(5x) \cdot 5
\end{align*}
8.
\(y(x) = 12(5.3^{x})\)
Solution.
\begin{align*}
y'(x) \amp = 12\ln(5.3)\cdot 5.3^x
\end{align*}