What is (\partial)/(\partial x)(1/(1-x/a-y/b)) ?

The answer to (\partial)/(\partial x)(1/(1-x/a-y/b)) is (ab^2)/((ab-xb-ay)^2)

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(\partial)/(\partial x)(1/(1-x/a-y/b))

\frac{\partial}{\partial x} (\frac{1}{1 - \frac{x}{a} - \frac{y}{b}})

\frac{a b ^{2}}{( ab - x b - a y ) ^{2}}

Solution steps

\frac{\partial}{\partial x} (\frac{1}{1 - \frac{x}{a} - \frac{y}{b}})

Treat

a, y, b

as constants

Apply exponent rule:

\frac{1}{a} = a^{- 1}

= \frac{\partial}{\partial x} ((1 - \frac{x}{a} - \frac{y}{b})^{- 1})

Apply the chain rule:

- \frac{1}{( 1 - \frac{x}{a} - \frac{y}{b} ) ^{2}} \frac{\partial}{\partial x} (1 - \frac{x}{a} - \frac{y}{b})

= - \frac{1}{( 1 - \frac{x}{a} - \frac{y}{b} ) ^{2}} \frac{\partial}{\partial x} (1 - \frac{x}{a} - \frac{y}{b})

\frac{\partial}{\partial x} (1 - \frac{x}{a} - \frac{y}{b}) = - \frac{1}{a}

= - \frac{1}{( 1 - \frac{x}{a} - \frac{y}{b} ) ^{2}} (- \frac{1}{a})

Simplify

- \frac{1}{( 1 - \frac{x}{a} - \frac{y}{b} ) ^{2}} (- \frac{1}{a}) : \frac{a b ^{2}}{( ab - x b - a y ) ^{2}}

= \frac{a b ^{2}}{( ab - x b - a y ) ^{2}}

\int \frac{1}{12} x d x

sin (0.5 x)

y^{^{'}} = x + 8 y

\frac{d y}{d t} - y = 1, y (0) = 1

\int \frac{1}{25 - 4 x ^{2}} d x

What is (\partial)/(\partial x)(1/(1-x/a-y/b)) ?
The answer to (\partial)/(\partial x)(1/(1-x/a-y/b)) is (ab^2)/((ab-xb-ay)^2)