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Differential Calculus Formulas

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Rules

1. Sum Rule:

   ${\left[f\right(x)\pm g(x\left)\right]}^{\prime }=f^{\prime }\left(x\right)\pm g^{\prime }\left(x\right)$

2. Product Rule:

   ${\left[f\right(x)·g(x\left)\right]}^{\prime }=f^{\prime }\left(x\right)·g\left(x\right)+f\left(x\right)·g^{\prime }\left(x\right)$

3. Quotient Rule:

   ${\left[\frac{f\left(x\right)}{g\left(x\right)}\right]}^{\prime }=\frac{f^{\prime }\left(x\right)·g\left(x\right)-f\left(x\right)·g^{\prime }\left(x\right)}{g^2\left(x\right)}$

4. Chain Rule:

   ${[f\circ g(x\left)\right]}^{\prime }=f^{\prime }\left(g\right(x\left)\right)·g^{\prime }\left(x\right)$

5. Inverse Rule:

   ${\left[f^{-1}\right]}^{\prime }(t)=\frac{1}{f^{\prime }(f^{-1}(t))}$

Reference Formulas:

6. $\frac{d}{dx}\mathrm{c}=0$
7. $\frac{d}{dx}\mathrm{c}f\left(x\right)=\mathrm{c}\frac{d}{dx}f\left(x\right)$
8. $\frac{d}{dx}{x}^n=n{x}^{n-1}$
9. $\frac{d}{dx}\sin ax=a\cos ax$
10. $\frac{d}{dx}\cos ax=-a\sin ax$
11. $\frac{d}{dx}\tan x={\sec }^{2}x$
12. $\frac{d}{dx}\cot x=-{\csc }^{2}x$
13. $\frac{d}{dx}\sec x=\sec x \tan x$
14. $\frac{d}{dx}\csc x=-\csc x \cot x$
15. $\frac{d}{dx}{e}^{ax}={ae}^x$
16. $\frac{d}{dx}{a}^x={\ln \left(a\right) a}^x, a>0 \& a\ne 1$
17. $\frac{d}{dx}\ln x=\frac{1}{x}$
18. $\frac{d}{dx}{\log }_a\left(x\right)=\frac{1}{\ln \left(a\right) x}, a>0 \&a\ne 1$
19. $\frac{d}{dx}\arcsin x=\frac{1}{\sqrt{1-x^2}}$
20. $\frac{d}{dx}\arccos x=-\frac{1}{\sqrt{1-x^2}}$
21. $\frac{d}{dx}\arctan x=\frac{1}{1+x^2}$

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