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# Differentiation of functions

Differentiation is the action of computing a derivative. The derivative of a function y = f(x) of a variable x is a measure of the rate at which the value y of the function changes with respect to the change of the variable x. It is called the derivative of f with respect to x

Summary of results discussed so far at a glance

1. $\displaystyle \frac { d }{ dx } \left( { x }^{ n } \right) =n{ x }^{ n-1 }$
2. $\displaystyle \frac { d }{ dx } \left( { e }^{ x } \right) ={ e }^{ x }$
3. $\displaystyle \frac { d }{ dx } \left( { a }^{ x } \right) ={ a }^{ x }\log _{ e }{ a }$
4. $\displaystyle \frac { d }{ dx } \left( \log _{ e }{ x } \right) =\frac { 1 }{ x }$
5. $\displaystyle \frac { d }{ dx } \left( \log _{ a }{ x } \right) =\frac { 1 }{ x\log _{ e }{ a } }$
6. $\displaystyle \frac { d }{ dx } \left( \sin { x } \right) =\cos { x }$
7. $\displaystyle \frac { d }{ dx } \left( \cos { x } \right) =-\sin { x }$
8. $\displaystyle \frac { d }{ dx } \left( \tan { x } \right) =\sec ^{ 2 }{ x }$
9. $\displaystyle \frac { d }{ dx } \left( \cot { x } \right) =-{ cosec }^{ 2 }x$
10. $\displaystyle \frac { d }{ dx } \left( \sec { x } \right) =\sec { x } \tan { x }$
11. $\displaystyle \frac { d }{ dx } \left( cosec\quad x \right) =-cosec\quad x\quad \cot { x }$
12. $\displaystyle \frac { d }{ dx } \left( \sin ^{ -1 }{ x } \right) =\frac { 1 }{ \sqrt { 1-{ x }^{ 2 } } }$
13. $\displaystyle \frac { d }{ dx } \left( \cos ^{ -1 }{ x } \right) =-\frac { 1 }{ \sqrt { 1-{ x }^{ 2 } } }$
14. $\displaystyle \frac { d }{ dx } \left( \tan ^{ -1 }{ x } \right) =\frac { 1 }{ 1+{ x }^{ 2 } }$
15. $\displaystyle \frac { d }{ dx } \left( \cot ^{ -1 }{ x } \right) =-\frac { 1 }{ 1+{ x }^{ 2 } }$
16. $\displaystyle \frac { d }{ dx } \left( \sec ^{ -1 }{ x } \right) =\frac { 1 }{ \left| x \right| \sqrt { { x }^{ 2 }-1 } }$
17. $\displaystyle \frac { d }{ dx } \left( { cosec }^{ -1 }x \right) =-\frac { 1 }{ \left| x \right| \sqrt { { x }^{ 2 }-1 } }$
April 18, 2019
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