What is the derivative of (x-2)(1/(x+2))(x^3) ?

The derivative of (x-2)(1/(x+2))(x^3) is (3x^4+4x^3-12x^2)/((x+2)^2)

What is the first derivative of (x-2)(1/(x+2))(x^3) ?

The first derivative of (x-2)(1/(x+2))(x^3) is (3x^4+4x^3-12x^2)/((x+2)^2)

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derivative

\frac{3 x ^{4} + 4 x ^{3} - 12 x ^{2}}{( x + 2 ) ^{2}}

Solution steps

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

Apply the Product Rule:

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

f = x - 2, g = \frac{1}{x + 2} x^{3}

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

\frac{d}{d x} (x - 2) = 1

\frac{d}{d x} (\frac{1}{x + 2} x^{3}) = \frac{2 x ^{3} + 6 x ^{2}}{( x + 2 ) ^{2}}

= 1 \cdot \frac{1}{x + 2} x^{3} + \frac{2 x ^{3} + 6 x ^{2}}{( x + 2 ) ^{2}} (x - 2)

Simplify

1 \cdot \frac{1}{x + 2} x^{3} + \frac{2 x ^{3} + 6 x ^{2}}{( x + 2 ) ^{2}} (x - 2) : \frac{3 x ^{4} + 4 x ^{3} - 12 x ^{2}}{( x + 2 ) ^{2}}

= \frac{3 x ^{4} + 4 x ^{3} - 12 x ^{2}}{( x + 2 ) ^{2}}

What is the derivative of (x-2)(1/(x+2))(x^3) ?
The derivative of (x-2)(1/(x+2))(x^3) is (3x^4+4x^3-12x^2)/((x+2)^2)
What is the first derivative of (x-2)(1/(x+2))(x^3) ?
The first derivative of (x-2)(1/(x+2))(x^3) is (3x^4+4x^3-12x^2)/((x+2)^2)