What is the derivative of x|x-2| ?

The derivative of x|x-2| is (2x^2-6x+4)/(|x-2|)

What is the first derivative of x|x-2| ?

The first derivative of x|x-2| is (2x^2-6x+4)/(|x-2|)

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derivative of

x ∣ x - 2 ∣

\frac{2 x ^{2} - 6 x + 4}{∣ x - 2 ∣}

Solution steps

\frac{d}{d x} (x ∣ x - 2 ∣)

Apply the Product Rule:

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

f = x, g = ∣ x - 2 ∣

= \frac{d x}{d x} ∣ x - 2 ∣ + \frac{d}{d x} (∣ x - 2 ∣) x

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

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

= 1 \cdot ∣ x - 2 ∣ + \frac{x - 2}{∣ x - 2 ∣} x

Simplify

1 \cdot ∣ x - 2 ∣ + \frac{x - 2}{∣ x - 2 ∣} x : \frac{2 x ^{2} - 6 x + 4}{∣ x - 2 ∣}

= \frac{2 x ^{2} - 6 x + 4}{∣ x - 2 ∣}

x \to - 2 lim (2 ∣ x + 2 ∣)

\frac{\partial}{\partial x} (ln (y + x))

\int_{- 2}^{4} x d x

\int \frac{1}{1 - 3 u} d u

\frac{\partial}{\partial x} (4 x^{2} + y^{2} - 9 y)

What is the derivative of x|x-2| ?
The derivative of x|x-2| is (2x^2-6x+4)/(|x-2|)
What is the first derivative of x|x-2| ?
The first derivative of x|x-2| is (2x^2-6x+4)/(|x-2|)