Question

Find the derivatives w.r.t.x:
$\rm \frac{1}{\cos x}+3^{x+5}-\frac{4}{\log_{x} 5}$

Answer 1

we know that
$\frac{1}{cos \: x} = sec \: x \\ \\ log_{x}(5) = \frac{log \: 5}{log \: x}$
So for derivatives

$\frac{d(sec \: x)}{dx} = sec \: x \: tan \: x \\ \\ \frac{d( {3}^{(x + 5))} }{dx} = {3}^{(x + 5)} log \: 3 \: \: (1) \\ \\ \frac{d}{dx} ( \frac{log \:x }{log \: 5} ) \\ = ( \frac{1 }{log \: 5} ) \frac{d}{dx} (log\:x) = = \frac{1}{x(log \: 5) }$
put all these derivatives in one expression for final answer

$\frac{d}{dx} \bigg(\frac{1}{\cos x}+3^{x+5}-\frac{4}{\log_{x} 5}\bigg) \\\\= sec \: x \: tan \: x + {3}^{(x + 5)} log \: 3 - \frac{4}{x(log \: 5) }$
Hope it helps you.