Answer to Question #124049 in Calculus for Nii Laryea

Question #124049
Differentiate the following functions with respect to x:
(i) ln(1 + sin² x)
(ii) x ͯ
1
Expert's answer
2020-07-01T19:05:51-0400

i. ln[1+sin2x]ln[1+sin^2x]

Differentiating with respect to x:x:

d[ln[1+sin2x]]/dxd[ln[1+sin^2x]]/dx

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

== (1/(1+sin2x))[d[1+(1cos2x)/2]/dx](1/(1+sin^2x))*[d[1+(1-cos2x)/2]/dx]

=(1/(1+sin2x))[d(2+1cos2x)/2]/dx]=(1/(1+sin^2x))*[d(2+1-cos2x)/2]/dx]

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

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


ii. xxx^x

Let A=xxx^x

Taking lnln on both sides:

lnA=xlnxln A=xlnx

Differentiating w.r.t xx :

d[lnA]/dx=d[xlnx]/dxd[ln A]/dx=d[xlnx]/dx

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

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

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


Need a fast expert's response?

Submit order

and get a quick answer at the best price

for any assignment or question with DETAILED EXPLANATIONS!

Comments

No comments. Be the first!

Leave a comment