3/13 unit 5 derivative & integral rules

derivative of an inverse function

(f⁻¹)’(x) = 1/f’(f⁻¹(x))

derivative of e^x

e^x

derivative of e^u

e^u * du/dx

∫e^xdx

e^x + C

∫e^udx

e^u + C

d/dx[a^x]

(ln a)(a^x)

d/dx[a^u]

(ln a)(a^u)du/dx

d/dx[log_ax]

1/(ln a * x)

d/dx[log_au]

1/(ln a * u) * du/dx

∫a^x dx

(1/ln a)a^x + C

d/dx[ln x]

1/x

d/dx[ln u]

1/u * du/dx, = u’/u

d/dx[ln |u|]

u’/u

∫1/x * dx

ln |x| + c

∫1/u * du

ln|u| + c

∫u’/u * dx

ln|u| + C

∫sin u du

-cos u + C

∫tan u du

-ln|cos u| + C

∫sec u du

ln|sec u + tan u| + C

∫cos u du

sin u + C

∫cot u du

ln|sin u| + C

∫csc u du

-ln|csc u + cot u| + C

d/dx[arcsin u]

u’/(√1-u²)

d/dx[arccos u]

-u’/(√1-u²)

d/dx[arctan u]

u’/(1 + u²)

d/dx[arccot u]

-u’/(1 + u²)

d/dx[arcsec u]

u’/(|u| * (√u² - 1))

d/dx[arccsc u]

-u’/(|u| * (√u² - 1))

∫du/(√a²-u²)

arcsin u/a + c

-∫du/(√a²-u²)

arccos u/a + c

∫du/(a² + u²)

(1/a arctan u/a) + c

-∫du/(a² + u²)

(1/a arccot u/a) + c

∫du/(u*√u²-a²)

(1/a arcsec |u|/a) + c

-∫du/(u*√u²-a²)

(1/a arccsc |u|/a) + c