What is (\partial)/(\partial x)(A^3(x/y)^2(1-x)) ?

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

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(\partial)/(\partial x)(A^3(x/y)^2(1-x))

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

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

Solution steps

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

Treat

A, y

as a constant

Take the constant out:

(a \cdot f)^{'} = a \cdot f^{'}

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

Apply the Product Rule:

(f \cdot g)^{'} = f^{'} \cdot g + f \cdot g^{'}

f = (\frac{x}{y})^{2}, g = 1 - x

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

\frac{\partial}{\partial x} ((\frac{x}{y})^{2}) = \frac{2 x}{y ^{2}}

\frac{\partial}{\partial x} (1 - x) = - 1

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

Simplify

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

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

\frac{d}{d x} (\frac{x ^{2} - 9}{x - 3})

\frac{d}{d x} (e^{x} (- x^{\frac{2}{3}} - \frac{4}{3} x^{- \frac{3}{2}}))

\int x^{2} (x^{3} + 2)^{4} d x

\int_{0}^{1} x (1 - x)^{2} d x

\int t e^{3 t} d t

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