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Table of Derivatives
Throughout this table, a and b are constants, independent of x.
F (x) F ′(x) = dF dx af (x) + bg(x) af ′(x) + bg′(x) f (x) + g(x) f ′(x) + g′(x) f (x) − g(x) f ′(x) − g′(x) af (x) af ′(x) f (x)g(x) f ′(x)g(x) + f (x)g′(x) f (x)g(x)h(x) f ′(x)g(x)h(x) + f (x)g′(x)h(x) + f (x)g(x)h′(x) f g ((xx)) f ′(x)g(x g)(−xf) 2 (x)g′(x) g(^1 x) −^ gg(′(xx))^2 f (g(x))^ f ′(g(x))g′(x) 1 0 a 0 xa^ axa−^1 g(x)a^ ag(x)a−^1 g′(x) sin x cos x sin g(x) g′(x) cos g(x) cos x − sin x cos g(x) −g′(x) sin g(x) tan x sec^2 x csc x − csc x cot x sec x sec x tan x cot x − csc^2 x ex^ ex eg(x)^ g′(x)eg(x) ax^ (ln a) ax ln x (^1) x ln g(x) g g′((xx)) loga x (^) x ln^1 a arcsin x √ 11 −x 2 arcsin g(x) √ 1 g−′(gx()x) 2 arccos x − √ 11 −x 2 arctan x (^) 1+^1 x 2 arctan g(x) (^) 1+g′g((xx)) 2 arccsc x − (^) x√ 11 −x 2 arcsec x (^) x√ 11 −x 2 arccot x − (^) 1+^1 x 2
Table of Indefinite Integrals
Throughout this table, a and b are given constants, independent of x and C is an arbitrary constant. f (x) F (x) = ∫^ f (x) dx af (x) + bg(x) a ∫^ f (x) dx + b ∫^ g(x) dx + C f (x) + g(x) ∫^ f (x) dx + ∫^ g(x) dx + C f (x) − g(x) ∫^ f (x) dx − ∫^ g(x) dx + C af (x) a ∫^ f (x) dx + C u(x)v′(x) u(x)v(x) − ∫^ u′(x)v(x) dx + C f (y(x))y′(x) F (y(x))^ where F (y) = ∫^ f (y) dy 1 x + C a ax + C xa^ x aa+1+1 + C if a 6 = − 1 (^1) x ln |x| + C g(x)ag′(x) g(x a+1)a+1 + C if a 6 = − 1 sin x − cos x + C g′(x) sin g(x) − cos g(x) + C cos x sin x + C tan x ln | sec x| + C csc x ln | csc x − cot x| + C sec x ln | sec x + tan x| + C cot x ln | sin x| + C sec^2 x tan x + C csc^2 x − cot x + C sec x tan x sec x + C csc x cot x − csc x + C ex^ ex^ + C eg(x)g′(x) eg(x)^ + C eax^1 a eax^ + C ax^ ln^1 a ax^ + C ln x x ln x − x + C √ 11 −x 2 arcsin x + C √^ g′(x) 1 −g(x)^2 arcsin^ g(x) +^ C √a (^21) −x 2 arcsin x a + C 1+^1 x^2 arctan^ x^ +^ C 1+^ g′g((xx))^2 arctan^ g(x) +^ C a^2 +^1 x^21 a arctan^ x a +^ C x√ 11 −x^2 arcsec^ x^ +^ C
Properties of Logarithms
In the following, x and y are arbitrary real numbers that are strictly bigger than 0, a is an arbitrary constant that is strictly bigger than one and e is 2.7182818284, to ten decimal places.
- eln^ x^ = x, aloga^ x^ = x, loge x = ln x, loga x = ln ln^ xa
- loga^ (ax)^ = x, ln (ex)^ = x ln 1 = 0, loga 1 = 0 ln e = 1, loga a = 1
- ln(xy) = ln x + ln y, loga(xy) = loga x + loga y
- ln (^ x y^ )^ = ln x − ln y, loga^ (^ x y^ )^ = loga x − loga y ln (^1 y^ )^ = − ln y, loga^ (^ y^1 )^ = − loga y,
- ln(xy) = y ln x, loga(xy) = y loga x
- d dx ln x = (^) x^1 , d dx ln(g(x)) = g g′((xx)) , d dx loga x = (^) x ln^1 a
- ∫^ x^1 dx = ln |x| + C, ∫^ ln x dx = x ln x − x + C
- (^) xlim→∞ ln x = ∞, lim x→ 0 ln x = −∞ x^ lim→∞ loga^ x^ =^ ∞, lim x→ 0 loga^ x^ =^ −∞
- The graph of ln x is given below. The graph of loga x, for any a > 1, is similar.
x
y
y = ln x
