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미분공식

General Formulas

  1. $$ \frac{d}{dx}(c)=0 $$
  2. $$ \frac{d}{dx}[cf(x)]=cf \acute \ (x) $$
  3. $$ \frac{d}{dx}[f(x)+g(x)]=f \acute \ (x)+g \acute \ (x) $$
  4. $$ \frac{d}{dx}[f(x)-g(x)]=f \acute \ (x)-g \acute \ (x) $$
  5. $$ \frac{d}{dx}[f(x)g(x)]=f(x)g \acute \ (x)+g(x)f \acute \ (x) $$
  6. $$ \frac{d}{dx}\left[ \frac{f(x)}{g(x)} \right] =\frac{g(x)f \acute \ (x)-f(x)g \acute \ (x)}{[g(x)]^{2}} $$
  7. $$ \frac{d}{dx}f(g(x))=f \acute \ (g(x))g \acute \ (x) $$
  8. $$ \frac{d}{dx}(x^{n})=nx^{n-1} $$

Exponential And Logarithmic Functions

  1. $$ \frac{d}{dx}(e^{x})=e^{x} $$
  2. $$ \frac{d}{dx}(a^{x})=a^{x}ln \ a $$
  3. $$ \frac{d}{dx}ln|x|=\frac{1}{x} $$
  4. $$ \frac{d}{dx}(log_{a}x)=\frac{1}{x \ ln \ a} $$

Trigonometric Functions

  1. $$ \frac{d}{dx}(sin \ x)=cos \ x $$
  2. $$ \frac{d}{dx}(cos \ x)=-sin \ x $$
  3. $$ \frac{d}{dx}(tan \ x)=sec^{2} \ x $$
  4. $$ \frac{d}{dx}(csc \ x)=-csc \ x \ cot \ x $$
  5. $$ \frac{d}{dx}(sec \ x)=sec \ x \ tan \ x $$
  6. $$ \frac{d}{dx}(cot \ x)=-csc^{2} \ x $$

Inverse Trigonometric Functions

  1. $$ \frac{d}{dx}(sin^{-1} \ x)=\frac{1}{\sqrt{1-x^{2}}} $$
  2. $$ \frac{d}{dx}(cos^{-1} \ x)=-\frac{1}{\sqrt{1-x^{2}}} $$
  3. $$ \frac{d}{dx}(tan^{-1} \ x)=\frac{1}{1+x^{2}} $$
  4. $$ \frac{d}{dx}(csc^{-1} \ x)=-\frac{1}{x \sqrt{x^{2}-1}} $$
  5. $$ \frac{d}{dx}(sec^{-1} \ x)=\frac{1}{x \sqrt{x^{2}-1}} $$
  6. $$ \frac{d}{dx}(cot^{-1} \ x)=-\frac{1}{1+x^{2}} $$

Hyperbolic Functions

  1. $$ \frac{d}{dx}(sinh \ x)=cosh \ x $$
  2. $$ \frac{d}{dx}(cosh \ x)=sinh \ x $$
  3. $$ \frac{d}{dx}(tanh \ x)=sech^{2} \ x $$
  4. $$ \frac{d}{dx}(csch \ x)=-csch \ x \ coth \ x $$
  5. $$ \frac{d}{dx}(sech \ x)=-sech \ x \ tanh \ x $$
  6. $$ \frac{d}{dx}(coth \ x)=-csch^{2} \ x $$

Inverse Hyperbolic Functions

  1. $$ \frac{d}{dx}(sinh^{-1} \ x)=\frac{1}{\sqrt{1+x^{2}}} $$
  2. $$ \frac{d}{dx}(cosh^{-1} \ x)=\frac{1}{\sqrt{x^{2}-1}} $$
  3. $$ \frac{d}{dx}(tanh^{-1} \ x)=\frac{1}{1-x^{2}} $$
  4. $$ \frac{d}{dx}(csch^{-1} \ x)=-\frac{1}{|x| \ \sqrt{x^{2}+1}} $$
  5. $$ \frac{d}{dx}(sech^{-1} \ x)=-\frac{1}{x\sqrt{1-x^{2}}} $$
  6. $$ \frac{d}{dx}(coth^{-1} \ x)=\frac{1}{1-x^{2}} $$