| \( c \) |
\( 0 \) |
|
|
| \( x^p \) |
\( px^{p-1} \) |
\( \left[ u(x) \right]^p \) |
\( p\left[ u(x) \right]^{p-1} u'(x) \) |
| \( \displaystyle\sqrt x \) |
\( \displaystyle\frac{1}{2 \sqrt x} \) |
\( \displaystyle\sqrt {u(x)} \) |
\( \displaystyle\frac{u'(x)}{2 \sqrt {u(x)}} \) |
| \( \displaystyle\frac 1 x \) |
\( \displaystyle-\frac 1{x^2} \) |
\( \displaystyle\frac {1}{u(x)} \) |
\( \displaystyle-\frac {u'(x)}{\left[ u(x) \right]^2} \) |
| \( \displaystyle\frac 1 {x^2} \) |
\( \displaystyle-\frac 2{x^3} \) |
\( \displaystyle\frac 1 {\left[ u(x) \right]^2} \) |
\( \displaystyle-\frac {2u'(x)}{\left[ u(x) \right]^3} \) |
| \( \displaystyle \sin x \) |
\( \displaystyle \cos x \) |
\( \displaystyle \sin {\left[ u(x) \right]} \) |
\( \displaystyle \cos {\left[ u(x) \right] \cdot u'(x)} \) |
| \( \displaystyle \cos x \) |
\( \displaystyle -\sin x \) |
\( \displaystyle \cos {\left[ u(x) \right]} \) |
\( \displaystyle -\sin {\left[ u(x) \right] \cdot u'(x)} \) |
| \( \displaystyle \tan x \) |
\( \displaystyle \frac{1}{\cos^2 x} \) |
\( \displaystyle \tan {\left[ u(x) \right]} \) |
\( \displaystyle \frac{u'(x)}{\cos^2 \left[ u(x) \right]}\) |
| \( \displaystyle \ln x \) |
\( \displaystyle \frac{1}{x} \) |
\( \displaystyle \ln {\left[ u(x) \right]} \) |
\( \displaystyle \frac{u'(x)}{u(x)} \) |
| \( \displaystyle \log_a x \) |
\( \displaystyle \frac{1}{x \cdot \ln a } \) |
\( \displaystyle \log_a {\left[ u(x) \right]} \) |
\( \displaystyle \frac{u'(x)}{u(x) \cdot \ln a} \) |
| \( \displaystyle \mathrm{e}^x \) |
\( \displaystyle \mathrm{e}^x \) |
\( \displaystyle \mathrm{e}^{u(x)} \) |
\( \displaystyle \mathrm{e}^{u(x)} \cdot u'(x) \) |
| \( \displaystyle a^x \) |
\( \displaystyle a^x \cdot \ln a \) |
\( \displaystyle a^{u(x)} \) |
\( \displaystyle a^{u(x)} \cdot u'(x) \cdot \ln a \) |
| \( \displaystyle \arcsin x \) |
\( \displaystyle \frac{1}{\sqrt{1-x^2}} \) |
\( \displaystyle \arcsin \left[u(x)\right] \) |
\( \displaystyle \frac{u'(x)}{\sqrt{1-\left[u(x)\right]^2}} \) |
| \( \displaystyle \arccos x \) |
\( \displaystyle \frac{-1}{\sqrt{1-x^2}} \) |
\( \displaystyle \arccos \left[u(x)\right] \) |
\( \displaystyle \frac{-u'(x)}{\sqrt{1-\left[u(x)\right]^2}} \) |
| \( \displaystyle \arctan x \) |
\( \displaystyle \frac{1}{1+x^2} \) |
\( \displaystyle \arctan \left[u(x)\right] \) |
\( \displaystyle \frac{u'(x)}{1+\left[u(x)\right]^2} \) |