limits-and-derivatives

Limits

• ${\lim }_{x\to 0}\frac{\sin (x)}{x}={\lim }_{x\to 0}\frac{1-\cos (x)}{x}={\lim }_{x\to 0}\frac{\tan (x)}{x}=1$
• ${\lim }_{x\to 0}\frac{{\sin }^{-1}(x)}{x}={\lim }_{x\to 0}\frac{{\tan }^{-1}(x)}{x}=1$
• ${\lim }_{x\to 0}\frac{e^x-1}{x}={\lim }_{x\to \infty }\frac{\ln (1+x)}{x}=1$
• ${\lim }_{x\to 0}(1+x{)}^{1/x}={\lim }_{x\to \infty }(1+\frac{1}{x}{)}^x=e$
• ${\lim }_{x\to 0}\frac{a^x-1}{x}=\ln (a)$
• ${\lim }_{x\to \infty }(1+\frac{a}{x}{)}^x=e^a$
• ${\lim }_{x\to a}\frac{x^n-a^n}{x-a}=n{a}^{n-1}$
• ${\lim }_{n\to \infty }\frac{t^n-1}{t^n+1}=1$
• ${\lim }_{x\to a}(1+f(x){)}^{g(x)}=e^{{\lim }_{x\to a}(f(x)\times g(x))}$

Derivatives

• $\frac{d}{dx}[f(g(x))]=f^{\prime }(g(x))\cdot g^{\prime }(x)$
• $\frac{d}{dx}[x^n]=n{x}^{n-1}$
• $\frac{d}{dx}[e^x]=e^x$
• $\frac{d}{dx}[e^{f(x)}]=f^{\prime }(x)\cdot e^{f(x)}$
• $\frac{d}{dx}[a^x]=a^x\ln (a)$
• $\frac{d}{dx}[\sin x]=\cos (x)$
• $\frac{d}{dx}[\cos x]=-\sin (x)$
• $\frac{d}{dx}[\tan x]={\sec }^2(x)$
• $\frac{d}{dx}[\cot x]=-{\csc }^2(x)$
• $\frac{d}{dx}[\sec x]=\sec (x)\tan (x)$
• $\frac{d}{dx}[\csc x]=-\csc (x)\cot (x)$
• $\frac{d}{dx}[\ln |x|]=\frac{1}{x}$
• $\frac{d}{dx}[{\sin }^{-1}x]=\frac{1}{\sqrt{1-x^2}}$
• $\frac{d}{dx}[{\cos }^{-1}x]=\frac{-1}{\sqrt{1-x^2}}$
• $\frac{d}{dx}[{\tan }^{-1}x]=\frac{1}{x^2+1}$
• $\frac{d}{dx}[{\cot }^{-1}x]=\frac{-1}{x^2+1}$
• $\frac{d}{dx}[{\sec }^{-1}x]=\frac{1}{|x|\sqrt{x^2-1}}$
• $\frac{d}{dx}[{\csc }^{-1}x]=\frac{-1}{|x|\sqrt{x^2-1}}$