derivative of f(x)=(-3x^3+3x^2)^3

We save the definition:

f (x) = {(- 3 x^{3} + 3 x^{2})}^{3}

Let's perform the differentiation:

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

We differentiate using:

\frac{d}{d x} (f^{n}) = n \cdot \frac{d}{d x} (f) \cdot f^{n - 1}

(where f =

- 3 x^{3} + 3 x^{2}

)

ⓘ The derivative of x^n is n*x^(n-1):

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

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

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

ⓘ The derivative of a sum is the sum of the derivatives:

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

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

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

ⓘ A constant factor stays in front of the derivative, (k*u)' = k*u':

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

-3 \cdot \frac{d}{d x} (x^{3})

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

3 \cdot \frac{d}{d x} (x^{2})

■The derivative of x^n is n*x^(n-1):

\frac{d}{d x} (x^{3})

3 x^{2}

\frac{d}{d x} (x^{2})

2 x

3 \cdot (- 3 \cdot 3 x^{2} + 3 \cdot 2 x) \cdot {(- 3 x^{3} + 3 x^{2})}^{2}

ⓘ Multiplications are done before additions and subtractions:

3 \cdot 3

9

3 \cdot 2

6

3 \cdot (- 9 x^{2} + 6 x) \cdot {(- 3 x^{3} + 3 x^{2})}^{2}

▶Derivative:

3 \cdot (6 x - 9 x^{2}) \cdot {(3 x^{2} - 3 x^{3})}^{2}