Fundamentals of Differentiation

$\frac{x}{\ln(x)}$

$-\frac{1}{x}$

$\ln x$

$\frac{1}{x}$

$h'(t)= -\cot(t)$

$h'(t)= -\sec(t)$

$h'(t)= -\tan(t)$

$h'(t)= -\csc(t)$

$-\\sin x$

$\\sin x$

$-\\cos x$

$\\cos x$

$k = 2$

$k = 3$

$k = \frac{3}{e}$

$k = \frac{3}{2}$

$\ln(x)$

$e^x$

$0$

$xe^x$

$-\\sin 2x$

$2 \\sin 2x$

$\\cos 2x$

$-2 \\sin 2x$

$\frac{dg}{dt} = \frac{1}{4}$

$\frac{dg}{dt} = \frac{1}{16}$

$\frac{dg}{dt} = 0$

$\frac{dg}{dt} = \frac{1}{8}$

$2x e^{2x}$

$2x$

$2e^{2x}$

$2$

-$\frac{1}{2}e^{-2x}$

-$4e^{-4x}$

-$2e^{-2x}$

-$e^{-2x}$

The function has a horizontal tangent line at that point.

The function's slope becomes negative.

The function has no extremum point.

The function's slope increases without bound.