Section 1.4 Practice Gateway Exam 4
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.
\(f(x) = x^{6} + \dfrac{1}{\sqrt{x}}-100^{\pi}\)
Solution.
\begin{align*}
f'(x) \amp = \frac{d}{dx}[x^6 + x^{-1/2} - 100^{\pi}]\\
\amp = 6x^5 - \frac{1}{2}x^{-3/2}
\end{align*}
2.
\(g(x) = \cos^4(x^2+x)\)
Solution.
\begin{align*}
g'(x) \amp = \frac{d}{dx}[(\cos(x^2+x))^4]\\
\amp = 4\cos^3(x^2+x) \cdot \frac{d}{dx}[\cos(x^2+x)]\\
\amp = 4\cos^3(x^2+x) \cdot -\sin(x^2+x) \cdot \frac{d}{dx}[x^2+x]\\
\amp = 4\cos^3(x^2+x) \cdot -\sin(x^2+x) \cdot (2x+1)
\end{align*}
3.
\(r(x) = 3\sin(\ln(x))\)
Solution.
\begin{align*}
r'(x) \amp = 3\cos(\ln(x))\cdot \frac{d}{dx}[\ln(x)]\\
\amp = 3\cos(\ln(x))\cdot \frac{1}{x}
\end{align*}
4.
\(y = \dfrac{x^3+x}{x^2}\)
Solution.
\begin{align*}
\frac{dy}{dx} \amp = \frac{d}{dx}[x+x^{-1}]\\
\amp = 1-x^{-2}
\end{align*}
5.
\(f(x) = 2.1e^{\sin(x^2+1)}\)
Solution.
\begin{align*}
f'(x) \amp = 2.1e^{\sin(x^2+1)}\cdot \frac{d}{dx}[\sin(x^2+1)]\\
\amp = 2.1e^{\sin(x^2+1)}\cdot \cos(x^2+1) \cdot \frac{d}{dx}[x^2+1]\\
\amp = 2.1e^{\sin(x^2+1)}\cdot \cos(x^2+1) \cdot 2x
\end{align*}
6.
\(g(\theta) = \tan(\theta)\sin(\theta)\)
Solution.
\begin{align*}
g'(\theta) \amp = \tan(\theta)\cdot \frac{d}{d\theta}[\sin(\theta)] + \sin(\theta) \cdot \frac{d}{d\theta}[\tan(\theta)]\\
\amp = \tan(\theta)\cdot \cos(\theta) +\sin(\theta) \cdot \sec^2(\theta)
\end{align*}
7.
\(h(t) = \dfrac{t^3}{\cos(3t)}\)
Solution.
\begin{align*}
h'(t) \amp = \frac{\cos(3t) \cdot \frac{d}{dt}[t^3] - t^3 \frac{d}{dt}[\cos(3t)]}{(\cos(3x))^2}\\
\amp = \frac{\cos(3t) \cdot 3t^2 - t^3 \cdot (-\sin(3t)) \cdot 3}{\cos^2(3t)}
\end{align*}
8.
\(h(x) = 5\sin(4x+1)\)
Solution.
\begin{align*}
h'(x) \amp = 5\cos(4x+1)\cdot \frac{d}{dx}[4x+1]\\
\amp = 5\cos(4x+1) \cdot 4
\end{align*}