Find the derivative of `y = ln x^3`

Answer

2

Explanation

a) `y=kx+k^(2)`
`rArr (dy)/(dx) =k`
Put the value of k
`rArr y=x(dy)/(dx) + ((dy)/(dx))^(2)`
b) `y=ae^(2x)+be^(3x)`
`rArr e^(-5x//2) y = ae^(-x//2)+be^(x//2)`
Differentiating w.r.t. x,
`-5/2e^(-5x//2)y+e^(-5x//2)(dy)/(dx) = 12[-ae^(-x//2)+be^(x//2)]`
Again differentiating w.r.t. x,
`25/4e^(-(5x)/(2)), y-5/2e^((-5x)/2)(dy)/(dx)`
`-5/2e^((-5x)/2) (dy)/(dx) + e^((-5x)/2)(d^(2)y)/(dx^(2))=1/4[ae^(x/2)+be^(x/2)]`
`rArr 25/4e^(-5x//2) y-5e^(-5x//2)(d^(2)y)/(dx^(2))=1/4e^(-5x//2)y`
`(d^(2)y)/(dx^(2))-5(dy)/(dx) + 25/4y-1/4y=0`
`(d^(2)y)/(dx^(2))-5(dy)/(dx)+6y=0`
c) `y^(2)=4a(x+a)`
`rArr yy^(')=2a`
Substitute the value of a in the equation of the curve.
d) `xy=ae^(x)+be^(-x)+x^(2)`
`rArr x(dy)/(dx) +y=ae^(x)+be^(-x)+2`
`rArr (d^(2)y)/(dx^(2)) + (2dy)/(dx)-xy+x^(2)=2`