Question

If x = a^m/b^n, them find the absolute error in x.​

Answer 1

it is given that, x = a^m/b^n

taking logarithm both sides ,

logx = log(a^m/b^n)

⇒logx = log(a^m) - log(b^n)

⇒logx = mlog(a) - nlog(b)

now differentiating both sides,

⇒dx/x = m (da/a) - n(db/b)

let dx is comparable value of x, then dx → ∆x

similarly, da → ∆a and db → ∆b

so, ∆x/x = m(∆a/a) - n(∆b/b)

but , error is considered maximum.

so, use positive sign in place of negative sign.

i.e., ∆x/x = m(∆a/a) + n(∆b/b)

or, ∆x = x[m(∆a/a) + n(∆b/b) ]

hence, absolute error in x is x[m(∆a/a) + n(∆b/b) ]