What is the derivative of x(1-x) ?

The derivative of x(1-x) is 1-2x

What is the first derivative of x(1-x) ?

The first derivative of x(1-x) is 1-2x

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

1 - 2 x

Solution steps

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

Apply the Product Rule:

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

f = x, g = 1 - x

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

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

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

= 1 \cdot (1 - x) + (- 1) x

Simplify

1 \cdot (1 - x) + (- 1) x : 1 - 2 x

= 1 - 2 x

\frac{d}{d t} (sin (t^{2}))

\int e^{- x} x d x

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sin^{2} (x) + cos (x)