Answer to Question #121735 in Calculus for moreen

Question

Use logarithmic differentiation to solve dy over dy, if y = xsquared square root 7x-14 over (1+x) to the power 4

Answer 1

y = $\frac { x²\sqrt {7x-14}} {(1+x)⁴}$

ln(y) = ln [ $\frac { x²\sqrt {7x-14}} {(1+x)⁴}$ ]

Using properties of logarithm

ln(y) = ln(x²) + ln $\sqrt {7x-14}$ - ln(1+x)⁴

=> ln(y) = 2 ln x + $\frac {1} {2}$ ln (7x-14) - 4ln(1+x)

Differentiating with respect to x

$\frac {1} {y}$ $\frac {dy} {dx}$ = $\frac {2} {x}$ + $\frac {7} {2(7x-14)}$ - $\frac {4} {1+x}$

So

$\frac {dy} {dx}$ = y [ $\frac {2} {x}$ + $\frac {7} {2(7x-14)}$ - $\frac {4} {1+x}$ ]

= $\frac { x²\sqrt {7x-14}} {(1+x)⁴}$ [ $\frac {2} {x}$ + $\frac {7} {2(7x-14)}$ - $\frac {4} {1+x}$ ]