Differential and integral calculus

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Differential calculus

$$(\mathit{uv})\text{'}=u\text{'}v+\mathit{uv}\text{'}$$

$$\left(\frac{u}{v}\right)\text{'}=\frac{u\text{'}v-\mathit{uv}\text{'}}{v^2}$$

$$\underset{x\to 0}{\lim }\frac{\sin x}{x}=1$$

$$(x^r)\text{'}={\mathit{rx}}^{r-1}$$

$$(\sin x)\text{'}=\cos x$$

$$(\cos x)\text{'}=-\sin x$$

$$(\tan x)\text{'}=\frac{1}{{\cos }^{2}x}$$

$$(e^x)\text{'}=e^x$$

$$(a^x)\text{'}=a^x\log a$$

$$(\log x)\text{'}=\frac{1}{x}$$

$$({\log }_{a}x)\text{'}=\frac{1}{x\log a}$$

Integral calculus

$$\int f(x)\mathit{dx}=\int f(g(t))g\text{'}(t)\mathit{dt}$$

$$\int u\text{'}v\mathit{dx}=\mathit{uv}-\int \mathit{uv}\text{'}\mathit{dx}$$

$$\int \frac{1}{x}\mathit{dx}=\log \left|x\right|+C$$

$$\begin{array}{c}\underset{a}{\overset{b}{\int }}f(g(x))g\text{'}(x)\mathit{dx}=\underset{\alpha }{\overset{\beta }{\int }}f(t)\mathit{dt}\\ \mathit{where}g(x)=t,g(a)=\alpha ,g(b)=\beta \end{array}$$

$$\underset{a}{\overset{b}{\int }}f(x)g\text{'}(x)\mathit{dx}={[f(x)g(x)]}_a^b-\underset{a}{\overset{b}{\int }}f\text{'}(x)g(x)\mathit{dx}$$

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