2 Pages Quiz 6A, Math 1850 Section 011
10-23-2014 Solutions Name
1. Find the derivative of ywith respect to the appropriate variable.(9)
(a) y= ln eθ
1 + eθ
Simplify y= ln eθ
−ln(1 + eθ) = θ−ln(1 + eθ) and now differentiate.
y′= 1 −
1
1 + eθeθ= 1 −
eθ
1 + eθ=1
1 + eθ
(b) y= 2(s2)
By the chain rule
y′= (ln 2)2s22s= (2 ln 2)s2s2
(because if f(x) = 2xthen f′(x) = (ln 2)2x).
(c) y= sin−1√2x
Differentiate, using the chain rule,
y′=1
q1−(√2x)2
√2 = √2
q1−(√2x)2
=r2
1−2x2
(d) y= tan−1(ln x)
Differentiate, using the chain rule,
y′=1
1 + (ln x)2
1
x=1
x(1 + (ln x)2)
2. Use logarithmic differentiation to find the derivative dy/dx if y=s(x+ 1)10
(2x+ 1)5
(4)
Take ln of both sides and simplify
ln y= ln s(x+ 1)10
(2x+ 1)5!
=1
2ln (x+ 1)10
(2x+ 1)5
=1
2(ln(x+ 1)10 −ln(2x+ 1)5) = 5 ln(x+ 1) −
5
2ln(2x+ 1)
Differentiate both sides in x
1
y
dy
dx =5
1 + x−
5
2
1
2x+ 1 2 = 5
1 + x−
5
2x+ 1
so that
dy
dx =5
1 + x−
5
2x+ 1y=5
1 + x−
5
2x+ 1s(x+ 1)10
(2x+ 1)5
