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Worksheet Basic Derivative Practice
Find the derivatives. For each function defined below, compute the derivative using the basic derivatives rules for power functions, exponential functions, sine and cosine, and sums and constant multiples.
1.
\(f(x) = x^3 + 3^x + 3^3\)
Answer .
\(f'(x) = 3x^2 + 3^x\ln(3)\)
2.
\(f(x) = 4\cos(x) - 7x^5\)
Answer .
\(f'(x) = -4\sin(x) - 35x^4\)
3.
\(f(x) = \frac{8}{x} + 2\sin(x)\)
Answer .
\(f'(x) = \frac{-8}{x^2} + 2 \cos(x)\)
4. Answer .
\(5x^4 + 9x^2\) (distribute first)
5.
\(\displaystyle f(x) = \sqrt{x} + \frac{1}{\sqrt{x}}\)
Answer .
\(f'(x) = \frac{1}{2\sqrt{x}} + -\frac{1}{2}x^{-3/2}\)
6.
\(f(x) = \sin(x) - \cos(x) + e^x + x^{e}\)
Answer .
\(f'(x) = \cos(x) + \sin(x) + e^x + ex^{e-1}\)
More derivatives. For each function defined below, compute the derivative using the basic derivatives rules for power functions, exponential functions, sine and cosine, and sums and constant multiples.
7. Answer .
\(P'(t) = 500(1.2^t)\ln(1.2)\)
8. Answer .
\(A'(x) = 2x + 10\) (multiply out the function before taking the derivative).
9.
\(\d h(t) = \frac{-9.8}{2}t^2 + 30 t + 100\)
10.
\(f(x) = e^t + e^{t+2} + e^2\)
Answer .
\(f'(x) = e^t + e^2e^t\) (note that
\(e^{t+2} = e^te^2\) )
11.
\(\d f(x) = 7\sqrt[5]{x} + \frac{7}{\sqrt[5]{x}}\)
Answer .
\(f'(x) = \frac{7}{5}x^{-4/5} + \frac{-7}{5}x^{-6/5}\)
12.
\(\d f(t) = \frac{t^2 + t + 1}{t^3}\)
Answer .
\(f'(t) = -t^{-2} - 2t^{-3} - 3t^{-4}\)
