# Differentiation Formulas – HSC Math

Posted In: Mathematics

In mathematics, differential calculus is a subfield of calculus concerned with the study of the rates at which quantities change. It is one of the two traditional divisions of calculus, the other being integral calculus, the study of the area beneath a curve.
if \$y=f(x)\$ is a function, then its first derivative, i.e. \${\dy}/{\dx}=f'(x)=\lim↙{h→0}{f(x+h)-f(x)}/h\$

General Formulas

1. \$\d/{\dx}(\c)=0,\$ where \$\c\$ is a constant.
2. \$\d/{\dx}(\c f(x))=\c\d/{\dx}(f(x))\$
3. \$\d/{\dx}(x^n\ )=nx^{n-1}\$
4. \$\d/{\dx}(sin\ x )=cos\ x\$
5. \$\d/{\dx}(cos\ x )=-sin\ x\$
6. \$\d/{\dx}(tan\ x )=sec^2\ x\$
7. \$\d/{\dx}(cot\ x )=-cosec^2\ x\$
8. \$\d/{\dx}(sec\ x )=sec\ x\ tan\ x\$
9. \$\d/{\dx}(cosec\ x )=-cosec\ x\ cot\ x\$
10. \$\d/{\dx}(e^x\ )=e^x\$
11. \$\d/{\dx}(e^{mx}\ )=me^{mx}\$
12. \$\d/{\dx}(ln\ x)=1/x\$
13. \$\d/{\dx}(a^x)=a^x\ ln\ a\$
14. \$\d/{\dx}(uv)=u\ \d/{\dx}(v)+v\ \d/{\dx}(u)\$
15. \$\d/{\dx}(u/v)={v\ \d/{\dx}(u)-u\ \d/{\dx}(v)}/{v\ ^2}\$
16. \$\d/{\dx}(sin^{-1}\ x )=1/{√{1-x^2}}\$
17. \$\d/{\dx}(cos^{-1}\ x )={-1}/{√{1-x^2}}\$
18. \$\d/{\dx}(tan^{-1}\ x )=1/{1+x^2}\$
19. \$\d/{\dx}(cot^{-1}\ x )={-1}/{1+x^2}\$
20. \$\d/{\dx}(sec^{-1}\ x )=1/{√{x(x^2-1)}\$
21. \$\d/{\dx}(cosec^{-1}\ x )={-1}/{√{x(x^2-1)}\$
22. \$\d/{\dx}(lox_a\ x )=1/x\ log_a\ e\$

Post Tags: Differentiation | HSC