What is the d/(dy)(ln(((5y+1)^2)/(sqrt(y^2+1)))) ?

The d/(dy)(ln(((5y+1)^2)/(sqrt(y^2+1)))) is (5y^2-y+10)/((5y+1)(y^2+1))

What is the first d/(dy)(ln(((5y+1)^2)/(sqrt(y^2+1)))) ?

The first d/(dy)(ln(((5y+1)^2)/(sqrt(y^2+1)))) is (5y^2-y+10)/((5y+1)(y^2+1))

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\frac{d}{d y} (ln (\frac{( 5 y + 1 ) ^{2}}{y ^{2} + 1}))

\frac{5 y ^{2} - y + 10}{( 5 y + 1 ) ( y ^{2} + 1 )}

Solution steps

\frac{d}{d y} (ln (\frac{( 5 y + 1 ) ^{2}}{y ^{2} + 1}))

Apply the chain rule:

\frac{1}{\frac{( 5 y + 1 ) ^{2}}{y ^{2} + 1}} \frac{d}{d y} (\frac{( 5 y + 1 ) ^{2}}{y ^{2} + 1})

= \frac{1}{\frac{( 5 y + 1 ) ^{2}}{y ^{2} + 1}} \frac{d}{d y} (\frac{( 5 y + 1 ) ^{2}}{y ^{2} + 1})

\frac{d}{d y} (\frac{( 5 y + 1 ) ^{2}}{y ^{2} + 1}) = \frac{25 y ^{3} + 49 y + 10}{( y ^{2} + 1 ) y ^{2} + 1}

= \frac{1}{\frac{( 5 y + 1 ) ^{2}}{y ^{2} + 1}} \cdot \frac{25 y ^{3} + 49 y + 10}{( y ^{2} + 1 ) y ^{2} + 1}

Simplify

\frac{1}{\frac{( 5 y + 1 ) ^{2}}{y ^{2} + 1}} \cdot \frac{25 y ^{3} + 49 y + 10}{( y ^{2} + 1 ) y ^{2} + 1} : \frac{5 y ^{2} - y + 10}{( 5 y + 1 ) ( y ^{2} + 1 )}

= \frac{5 y ^{2} - y + 10}{( 5 y + 1 ) ( y ^{2} + 1 )}

What is the d/(dy)(ln(((5y+1)^2)/(sqrt(y^2+1)))) ?
The d/(dy)(ln(((5y+1)^2)/(sqrt(y^2+1)))) is (5y^2-y+10)/((5y+1)(y^2+1))
What is the first d/(dy)(ln(((5y+1)^2)/(sqrt(y^2+1)))) ?
The first d/(dy)(ln(((5y+1)^2)/(sqrt(y^2+1)))) is (5y^2-y+10)/((5y+1)(y^2+1))