derival

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21 Terms

1
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(f(x)⋅g(x))′\left(f(x) \cdot g(x)\right)'

f′(x)⋅g(x)+f(x)⋅g′(x)f'(x) \cdot g(x) + f(x) \cdot g'(x)

2
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(f(x)g(x))′\left(\frac{f(x)}{g(x)}\right)'

f′(x)⋅g(x)−f(x)⋅g′(x)(g(x))2\frac{f'(x) \cdot g(x) - f(x) \cdot g'(x)}{(g(x))^2}

3
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(xn)′\left(x^n\right)'

n⋅xn−1n \cdot x^{n-1}

4
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(ax)′\left(a^x\right)'

ax⋅ln⁡(a)a^x \cdot \ln(a)

5
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(ex)′\left(e^x\right)'

exe^x

6
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(ln⁡(x))′\left(\ln(x)\right)'

1x\frac{1}{x}

7
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(sin⁡(x))′\left(\sin(x)\right)'

cos⁡(x)\cos(x)

8
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(cos⁡(x))′\left(\cos(x)\right)'

−sin⁡(x)-\sin(x)

9
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(tg(x))′\left(\text{tg}(x)\right)'

1cos⁡2(x)=1+tg2(x)\frac{1}{\cos^2(x)} = 1 + \text{tg}^2(x)

10
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(ctg(x))′\left(\text{ctg}(x)\right)'

−1sin⁡2(x)=−1−ctg2(x)\frac{-1}{\sin^2(x)} = -1 - \text{ctg}^2(x)

11
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(arcsin⁡(x))′\left(\arcsin(x)\right)'

11−x2\frac{1}{\sqrt{1-x^2}}

12
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(arccos⁡(x))′\left(\arccos(x)\right)'

−11−x2\frac{-1}{\sqrt{1-x^2}}

13
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(arctg(x))′\left(\text{arctg}(x)\right)'

11+x2\frac{1}{1+x^2}

14
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(arcctg(x))′\left(\text{arcctg}(x)\right)'

−11+x2\frac{-1}{1+x^2}

15
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x−nx^{-n}

1xn\frac{1}{x^n}

16
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(1x)′\left(\frac{1}{x}\right)'

(x−1)′=−1⋅x−2=−1x2\left(x^{-1}\right)' = -1 \cdot x^{-2} = \frac{-1}{x^2}

17
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xkn\sqrt[n]{x^k}

xknx^{\frac{k}{n}}

18
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1xkn\frac{1}{\sqrt[n]{x^k}}

x−knx^{-\frac{k}{n}}

19
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(x)′\left(\sqrt{x}\right)'

12x\frac{1}{2\sqrt{x}}

20
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(x3)′\left(\sqrt[3]{x}\right)'

13x23\frac{1}{3\sqrt[3]{x^2}}

21
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(xn)′\left(\sqrt[n]{x}\right)'

1nxn−1n\frac{1}{n\sqrt[n]{x^{n-1}}}