Answer to Question #124049 in Calculus for Nii Laryea

Question

Differentiate the following functions with respect to x:
(i) ln(1 + sin² x)
(ii) x ͯ

Answer 1

i. $ln[1+sin^2x]$

Differentiating with respect to $x:$

$d[ln[1+sin^2x]]/dx$

$=(1/(1+sin^2x))*[d(1+sin^2x)/dx]$

$=$ $(1/(1+sin^2x))*[d[1+(1-cos2x)/2]/dx]$

$=(1/(1+sin^2x))*[d(2+1-cos2x)/2]/dx]$

$=(1/2(1+sin^2x))*[2sin2x]$

$=sin2x/(1+sin^2x)$ (Answer)

ii. $x^x$

Let A=$x^x$

Taking $ln$ on both sides:

$ln A=xlnx$

Differentiating w.r.t $x$ :

$d[ln A]/dx=d[xlnx]/dx$

$=(1/A)*d[A]/dx=x*(1/x)+lnx$

or,$dA/dx=(1+lnx)*A$

$=(1+lnx)*x^x$ (Answer)