`F(x) = int_pi^ln(x) cos e^t dt` Find the derivative.

Saturday, February 2, 2013

$\displaystyle{F}{\left({x}\right)}={\int_{\pi}^{{{\ln{{x}}}}}}{\cos{{\left({{e}}^{{t}}\right)}}}{\left.{d}{t}\right.}$

$\displaystyle{F}'{\left({x}\right)}=?$

Take note that if the function has a form

$\displaystyle{F}{\left({x}\right)}={\int_{{a}}^{{{u}{\left({x}\right)}}}}{f{{\left({t}\right)}}}{\left.{d}{t}\right.}$

its derivative is

$\displaystyle{F}'{\left({x}\right)}={f{{\left({u}{\left({x}\right)}\right)}}}\cdot{u}'{\left({x}\right)}$

Applying this formula, the derivative of the function

$\displaystyle{F}{\left({x}\right)}={\int_{\pi}^{{{\ln{{\left({x}\right)}}}}}}{\cos{{\left({{e}}^{{t}}\right)}}}{\left.{d}{t}\right.}$

will be:

$\displaystyle{F}'{\left({x}\right)}={\cos{{\left({{e}}^{{{\ln{{\left({x}\right)}}}}}\right)}}}\cdot{\left({\ln{{\left({x}\right)}}}\right)}'$

$\displaystyle{F}'{\left({x}\right)}={\cos{{\left({{e}}^{{{\ln{{\left({x}\right)}}}}}\right)}}}\cdot\frac{{1}}{{x}}$

$\displaystyle{F}'{\left({x}\right)}={\cos{{\left({x}\right)}}}\cdot\frac{{1}}{{x}}$

$\displaystyle{F}'{\left({x}\right)}=\frac{{\cos{{\left({x}\right)}}}}{{x}}$

Therefore, the derivative of the given function is $\displaystyle{F}'{\left({x}\right)}=\frac{{\cos{{\left({x}\right)}}}}{{x}}$ .

Ninenotes