What is (\partial)/(\partial x)(1/(2x+3y)) ?

The answer to (\partial)/(\partial x)(1/(2x+3y)) is -2/((2x+3y)^2)

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(\partial)/(\partial x)(1/(2x+3y))

\frac{\partial}{\partial x} (\frac{1}{2 x + 3 y})

- \frac{2}{( 2 x + 3 y ) ^{2}}

Solution steps

\frac{\partial}{\partial x} (\frac{1}{2 x + 3 y})

Treat

y

as a constant

Apply exponent rule:

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

= \frac{\partial}{\partial x} ((2 x + 3 y)^{- 1})

Apply the chain rule:

- \frac{1}{( 2 x + 3 y ) ^{2}} \frac{\partial}{\partial x} (2 x + 3 y)

= - \frac{1}{( 2 x + 3 y ) ^{2}} \frac{\partial}{\partial x} (2 x + 3 y)

\frac{\partial}{\partial x} (2 x + 3 y) = 2

= - \frac{1}{( 2 x + 3 y ) ^{2}} \cdot 2

Simplify

- \frac{1}{( 2 x + 3 y ) ^{2}} \cdot 2 : - \frac{2}{( 2 x + 3 y ) ^{2}}

= - \frac{2}{( 2 x + 3 y ) ^{2}}

y = 3 - 2 x + 4 x^{3}, (- 1, 1)

\int 1 + cos (x) d x

x^{2} \frac{d y}{d x} - 2 x^{3} e^{- \frac{1}{x}} = y

\int sin (5 x) sin (4 x) d x

(4.1) (8.2)

What is (\partial)/(\partial x)(1/(2x+3y)) ?
The answer to (\partial)/(\partial x)(1/(2x+3y)) is -2/((2x+3y)^2)