Derivatives

15 flashcards · Mathematics · by Athena Papkouf

1. A function f:E→Ff : E → Ff:E→F is derivable in x0∈Ex_0 ∈ Ex0​∈E when

   lim⁡x→x0f(x)−f(x0)x−x0=lim⁡h→0f(x0+h)−f(x0)h:=f′(x0)\lim _{x \rightarrow x_{0}} \frac{f(x)-f\left(x_{0}\right)}{x-x_{0}}=\lim _{h \rightarrow 0} \frac{f\left(x_{0}+h\right)-f\left(x_{0}\right)}{h}:=f^{\prime}\left(x_{0}\right)limx→x0​​x−x0​f(x)−f(x0​)​=limh→0​hf(x0​+h)−f(x0​)​:=f′(x0​)

2. xrx^rxr

   rxr−1,r∈R,r≠0r x^{r-1}, r \in \mathbb{R}, r \neq 0rxr−1,r∈R,r̸​=0

3. loga(x)log_a(x)loga​(x)

   1xln(a)\frac{1}{xln(a)}xln(a)1​ 0, a\neq 1"> a>0,a≠1a>0, a\neq 1a>0,a̸​=1

4. exe^xex

   exe^xex

5. axa^xax

   0, a\neq 1"> axln(a),a>0,a≠1a^xln(a), a>0, a\neq 1axln(a),a>0,a̸​=1

6. log(x)log(x)log(x)

   1x\frac{1}{x}x1​

7. sin(x)sin(x)sin(x)

   cos(x)cos(x)cos(x)

8. cos(x)cos(x)cos(x)

   −sin(x)-sin(x)−sin(x)

9. tan(x)tan(x)tan(x)

   1cos2(x)\frac{1}{cos^2(x)}cos2(x)1​

10. arcsin(x)arcsin(x)arcsin(x)

    11−x2\frac{1}{\sqrt{1-x^{2}}}1−x2 ​1​

11. arccos(x)arccos(x)arccos(x)

    −11−x2-\frac{1}{\sqrt{1-x^{2}}}−1−x2 ​1​

12. arctg(x)arctg(x)arctg(x)

    11+x2\frac{1}{1+x^{2}}1+x21​

13. sinh(x)sinh(x)sinh(x)

    cosh(x)cosh(x)cosh(x)

14. cosh(x)cosh(x)cosh(x)

    sinh(x)sinh(x)sinh(x)

15. tanh(x)tanh(x)tanh(x)

    1cosh⁡2(x)\frac{1}{\operatorname{cosh}^{2}(x)}cosh2(x)1​