Assignment:
Inverse functions
Q1. Differentiate the problems:
1) f(x) = ln(x^2 + 10)
2) f(Ø) = ln(cos Ø)
3) f(x) =log2(1-3x)
4) f(x) = 5thROOT(ln x)
5) f(x)=SQRTx * (ln x)
6) f(t) = ln [(2t+1)^3 / (3t-1)^4]
7) h(x)=ln(x + SQRT(x^2-1))
8) g(x)=ln[(a-x)/(a+x)]
9) f(u)= (ln u) / (1+ln(2u))
10) h(t)=t^3-3^t
11) f(x)=cos(lnx)
12) f(x)=log10[(x)/(x-1)]
13) f(x)=ln 5thROOT(x)
14) f(x)= [(1+lnt)/(1-lnt)]
15) f(y)=yln(1+e^y)
16) y=ln(x^4sin2 x)
17) y=10^(tanØ)
18) y=ln|2-x-5x^2|
19) g(u)=ln[SQRT (3u+2/3u-2)]
20) y=ln(e^-x + xe^-x)
21) y=5^(-1/x)
22) y=2^3^(x^2)
Q2. Find y’ and y”
1) y=xlnx
2) y=log10x
3) y=(lnx)/x^2
4) y=ln(secx+tanx)
Q3. Differentiate f and find the domain of f.
1) f(x)= [(x)/(1-ln(x-1))]
2) f(x) = 1 / (1+lnx)
3) f(x)=x^2ln(1-x^2)
4) f(x)=ln ln ln x
Q4. Find f’(x)
1) f(x)=sinx+lnx
2) )f(x)=x^(cosx)
Q5. Use logarithmic differentiation to find the derivative of the function.
1) y=(2x+1)^5 (x^4-3)^6
2) y=(SQRTx)(e^x^2)(x^2+1)^10
3) y=[(sin2x)(tan4x)/(x^2 + 1)^2
4) y=x^x
5) y=x^(sinx)
6) y=(lnx)^x
7) y=x^e^x
8) y=x^(1/x)
9) y=(sinx)^x
Q6. Evaluate the integral
1) ∫(4 on top, 2 on bottom) (3/x) dx
2) ∫(2 on top, 1 on bottom) (dt) / (8-3t)
3) ∫(e on top, 1 on bottom) [(x^2+x+1)/ (x)] dx
4) ∫ [(2-x^2) /(6x-x^3)] dx
5) ∫(2 on top, 1 on bottom) [(4+u^2) / (u^3)] du
6) ∫(4 on top, 2 on bottom) (3/x) dx
7) ∫(9 on top, 4 on bottom) [(SQRTx) + (1/(SQRTX)]^2 dx
8) ∫(6 on top, e on bottom) [(dx)/ (xlnx)]
9) ∫ [(cosx) / (2+sinx)]dx
10) ∫(2 on top, 1 on bottom) (10t) dt
11) ∫ [(ln x)^2 / x] dx
12) ∫[(e^x) / (e^x+1)] dx
13)∫ [(x)(2^x^2) ] dx
Provide complete and step by step solution for the question and show calculations and use formulas.