- Find the derivatives of the following functions.
- [latex]f(x) = \frac{e^x}{x}[/latex]
[latex]f'(x) = \frac{x e^x - e^x}{x^2}[/latex]ans
- [latex]g(x) = \frac{\sqrt[3]{x}}{\ln(x)}[/latex]
[latex]g'(x) = \frac{\frac{1}{3}x^{-2/3}\ln(x) - \sqrt[3]{x}/x}{(\ln x)^2} = \frac{3x^{-2/3}(\ln(x) - 1)}{3 (\ln x)^2}[/latex]ans
- [latex]h(x) = \frac{1}{x^2 + 5x + 6}[/latex]
[latex]h'(x) = \frac{-(2x + 5)}{(x^2 + 5x + 6)^2}[/latex]ans
- [latex]i(x) = \frac{\cos(x)}{1 + x^2}[/latex]
[latex]i'(x) = \frac{-(1 + x^2) \sin(x) - 2x \cos(x)}{(1 + x^2)^2}[/latex]ans
- [latex]j(x) = \frac{\ln(x)}{x^2}[/latex]
[latex]j'(x) = \frac{1 - 2 \ln(x) }{x^3}[/latex]ans
- [latex]k(x) = e^{-x}[/latex] (How can you write this as a fraction?)
[latex]k'(x) = \frac{-e^x}{(e^x)^2} = -\frac{1}{e^x}[/latex]ans
- [latex]\ell(x) = \frac{x e^x}{1 + x}[/latex]
[latex]\ell'(x) = \frac{x^2 e^x + x e^x + e^x}{(1+x)^2}[/latex]ans
- [latex]f(x) = \frac{e^x}{x}[/latex]