What is the derivative of (a^2x/((x+b)^2)) ?

The derivative of (a^2x/((x+b)^2)) is (a^2(b-x))/((x+b)^3)

What is the first derivative of (a^2x/((x+b)^2)) ?

The first derivative of (a^2x/((x+b)^2)) is (a^2(b-x))/((x+b)^3)

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\frac{d}{d x} (\frac{a ^{2} x}{( x + b ) ^{2}})

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

Solution steps

\frac{d}{d x} (\frac{a ^{2} x}{( x + b ) ^{2}})

Take the constant out:

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

= a^{2} \frac{d}{d x} (\frac{x}{( x + b ) ^{2}})

Apply the Quotient Rule:

(\frac{f}{g})^{'} = \frac{f ^{'} \cdot g - g ^{'} \cdot f}{g ^{2}}

= a^{2} \frac{\frac{d x}{d x} ( x + b ) ^{2} - \frac{d}{d x} ( ( x + b ) ^{2} ) x}{( ( x + b ) ^{2} ) ^{2}}

\frac{d x}{d x} = 1

\frac{d}{d x} ((x + b)^{2}) = 2 (x + b)

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

Simplify

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

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

What is the derivative of (a^2x/((x+b)^2)) ?
The derivative of (a^2x/((x+b)^2)) is (a^2(b-x))/((x+b)^3)
What is the first derivative of (a^2x/((x+b)^2)) ?
The first derivative of (a^2x/((x+b)^2)) is (a^2(b-x))/((x+b)^3)