Derivatives and Antiderivatives to Remember

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44 Terms

1
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f’(x)
1
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f’(sinx)
cosx
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f’(cosx)
-sinx
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f’(tanx)
sec^2x
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f’(secx)
secxtanx
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f’(cscx)
-cscxcotx
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f’(cotx)
-csc^2x
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f’(a^x)
a^xlna
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f’(e^x)
e^x
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f’(lnx)
1/x
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f’(loga(x))
1/(xlna)
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f’(c)
0
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f’(x^n)
nx^n-1
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f’(f(g(x)))
f’(g(x)) \* g’(x)
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(f^-1)’(x)
1/(f’(f^-1(x))
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f’(sec^-1(x))
1/( |x| sqrt(x^2 -1))
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f’(csc^-1(x))
\- 1/( |x| sqrt(x^2 -1))
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f’(sin^-1(x))
1/ (sqrt(1- x^2))
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f’(cos^-1(x))
\- 1/ (sqrt(1- x^2))
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f’(tan^-1(x))
1 / (1+ x^2)
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f’(cot^-1(x))
\- 1 / (1+ x^2)
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(d/dx) (f(x)g(x))
f’(x)g(x) + f(x)g’(x)
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(d/dx) (f(x)/g(x))
f’(x)g(x) - f(x)g’(x) / g(x)^2
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ln 1
0
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e^0
1
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∫du
u + C
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∫u^n (du)
(u^n+1)/n+1 + C
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∫ 1/u (du)
ln |u| +C
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∫ a^u (du)
a^u (1/ ln a) + C
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∫ u dv
uv - ∫v du (Log Inverse trig Poly Trig Exp)
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Fundamental Theory of Calculas
(d/dx) ∫ f(x) dx = f(x)

∫ f(x) dx = F(b) - F(a) where F’(x) = f(x)
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2nd FTOC
(d/dx) ∫ f(x) dx = f(g(x)) g’(x)
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Mean Value Theorem
if a function f is *continuous* on the closed interval \[a,b\] and *differentiable* on the open interval (a,b), then there exists a point c in the interval (a,b) such that f'(c) is equal to the function's average rate of change over \[a,b\]: avg ROC = instant ROC at some point
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Intermediate value theorem
if f(x) is *continuous* and f(a)
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Extreme Value Theorem
if a function is *continuous* on a closed interval \[a,b\], then the function must have a maximum and a minimum on the interva
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∫ k f(x) dx
k ∫ f(x) dx
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∫ \[f(x) + g(x)\]dx
∫ f(x) dx + ∫ g(x) dx
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∫ (a to b) f(x) dx
\- ∫ (b to a) f(x) dx
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∫ (a to c) f(x) dx
∫ (a to b) f(x) dx + ∫ (b to c) f(x) dx
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∫ (a to a) f(x) dx
0
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f is integrable when
f is *continuous* over \[a,b\]

f is bounded on closed interval \[a,b\] and has at most a *finite* number of discontinuities
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selecting techniques
* long division when degree of deno
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ln |0|
undefined
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e ^x
exponential