Differentiate each of the following with respect to x.
Example 1 :
y = [sinxcos(x2)]/[x3 + lnx]
Solution :
Take logarithm on both sides
lny = ln[sinxcos(x2)]/[x3 + lnx]
lny = ln[sinxcos(x2)] - ln(x3 + lnx)
lny = ln(sinx) + lncos(x2) - ln(x3 + lnx) ----(1)
Derivative of ln y :
= (1/y)y'
= y'/y
Derivative of ln(sin x) :
= (1/sinx)cosx
= cosx/sinx
= cot x
Derivative of lncos (x2) :
= (1/cosx2) - sinx2(2x)
= -2xtan x2
Derivative of ln(x3 + ln x) :
= [1/(x3 + lnx)][3x2 + 1/x]
= [1/(x3 + lnx)][(3x3 + 1)/x]
= [(3x3 + 1)/x(x3 + lnx)]
(1) :
y'/y = cotx - 2xtanx2 - [(3x3 + 1)/x(x3 + lnx)]
ny' = y[cotx - 2xtanx2 - [(3x3 + 1)/x(x3 + lnx)]]
ny' = ([sinxcos(x2)]/[x3 + lnx])[cotx - 2xtanx2- [(3x3 + 1)/x(x3 + lnx)]]
Example 2 :
y = (x2 + 2)(x + √2)/√(x + 4)
Solution :
lny = ln{[(x2 + 2)(x + √2)]/√(x + 4)
lny = ln(x2 + 2) + ln(x + √2) - ln √(x + 4) ----(1)
Derivative of lny :
= (1/y)y'
= y'/y
Derivative of ln(x2 + 2) :
= (1/(x2 + 2))(2x)
= 2x/(x2+2)
Derivative of ln(x + √2) :
= 1/(x + √2)
Derivative of ln√(x + 4) :
= (1/√(x + 4)) ⋅ 1/2√(x + 4)
= 1/2(x + 4)
(1) :
y'/y = [2x/(x2 + 2)] + 1/(x + √2) + [1/2(x + 4)]
y' = y[2x/(x2 + 2)] + 1/(x + √2) + [1/2(x + 4)]
y' = ((x2 + 2)(x + √2)/√(x + 4))[2x/(x2 + 2)] + 1/(x + √2) +[1/2(x + 4)]
Example 3 :
y = x√x
Solution :
y = x√x
Take logarithm on both sides.
lny = ln(x√x)
lny = √xln(x)
Differentiate with respect to x.
u = √x and u' = 1/2√x
v = lnx and v' = 1/x
(1/y)y' = √x(1/x) + ln(x)(1/2√x)
= (√x/x) + lnx/2√x
= (1/√x) + lnx/2√x
(y'/y) = (2 + lnx)/2√x
y' = x√x(2 + lnx)/2√x
Example 4 :
y = xsinx
Solution :
y = xsinx
Take logarithm on both sides.
lny = lnxsinx
lny = sinxlnx
Differentiate with respect to x.
u = sinx and u' = cosx
v = lnx and v' = 1/x
(1/y)y' = sinx(1/x) + lnx(cosx)
y'/y = sinx/x + cosx(lnx)
y' = y[(sinx/x) + cosx(lnx)]
y' = xsinx[(sinx/x) + cosx(lnx)]
Example 5 :
y = (lnx)cosx
Solution :
y = (lnx)cosx
Take logarithm on both sides.
lny = ln((lnx)cosx)
lny = cosxln(lnx)
Differentiate with respect to x.
(1/y)y' = cosx(1/lnx)(1/x) + ln(lnx)(-sinx)
y' = y[cosx/xlnx) - sinx(ln (lnx)]
y' = (lnx)cosx(cosx/xlnx) - sinx(ln(lnx))
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