Function Derivatives &c

A table for convenience. Don't forget +c.
f(n)(x) f″(x) f′(x) f(x) ʃ f(x) dx
Generic Rules
$$f^{(n)}(x) + g^{(n)}(x)$$ $$f''(x) + g''(x)$$ $$f'(x) + g'(x)$$ $$f(x) + g(x)$$ $$\int f(x) + \int g(x)$$
$$...$$ $$f'(x)g(x) + f(x)g'(x)$$ $$f(x)g(x)$$ $$n/a$$
$$\frac {-f'(x)} {f(x)^2}$$ $$\frac 1 {f(x)}$$
$$\frac {f(x)g'(x)-g(x)f'(x)} {g(x)^2}$$ $$\frac {f(x)} {g(x)}$$
$$f'(g(x))g'(x)$$ $$f(g(x))$$
Functions
$$ax^{1-n}$$ $$0$$ $$a$$ $$ax$$ $$\frac a 2 x^2$$
$$ax^{b-n}\prod_{q=0}^{n-1} (b-q)$$ $$ab(b-1)x^{b-2}$$ $$abx^{b-1}$$ $$ax^{b}$$ $$\frac a b x^{b+1}$$
$$\frac 1 {x^n}$$ $$\frac 1 {x^2}$$ $$\frac 1 x$$ $$\ln(x)$$ $$x\ln(x)-x$$
$$\frac {(-1)^{n+1}} {x^n\ln(a)}$$ $$\frac {-1} {x^{2}\ln(a)}$$ $$\frac 1 {x\ln(a)}$$ $$\log_a (x)$$ $$x\log_a (\frac x e)$$
$$e^x$$ $$e^x$$ $$e^x$$ $$e^x$$ $$e^x$$
$$a^{x}\ln^n (a)$$ $$a^{x}\ln^2 (a)$$ $$a^{x}\ln (a)$$ $$a^{x}$$ $$\frac {a^x} {\ln (x)}$$
Trig
$$...$$ -sin(x) cos(x) sin(x) -cos(x)
2tan(x)sec2(x) sec2(x) tan(x) -ln(cos(x))
2cot(x)csc2(x) -csc2(x) cot(x) ln(sin(x))
sec(x)(tan2(x)+sec2(x)) tan(x)sec(x) sec(x) ln(tan(x)+sec(x))
csc(x)(cot2(x)+csc2(x)) -cot(x)csc(x) csc(x) -ln(cot(x)+csc(x))
x/(1-x2)3/2 1/sqrt(1-x2) arcsin(x) sqrt(1-x2)+xarcsin(x)
-x/(1-x2)3/2 -1/sqrt(1-x2) arccos(x) -sqrt(1-x2)+xarccos(x)
-2x/(x2+1)2 1/(x2+1) arctan(x) xarctan(x)-1/2*ln(x2+1)
2x/(x2+1)2 -1/(x2+1) arccot(x) xarccot(x)+1/2*ln(x2+1)
1/(x2sqrt(1-x-2)) arcsec(x)
-1/(x2sqrt(1-x-2)) arccsc(x)
Trig2
$$...$$ 2cos(2x) sin(2x) sin2(x) $$...$$
-2cos(2x) -2sin(x)cos(x) cos2(x)
-2sec4(x)(cos(2x)-2) 2tan(x)sec2(x) tan2(x)
2csc4(x)(cos(2x)+2) -2cot(x)csc2(x) cot2(x)
-2sec4(x)(cos(2x)-2) 2tan(x)sec2(x) sec2(x)
2csc4(x)(cos(2x)+2) -2cot(x)csc2(x) csc2(x)