derivative of x^2(1-x^2) — step-by-step solution

1) Let's perform the differentiation:

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

We differentiate

u1

x^{2}

and

v2

1 - x^{2}

using:

\frac{d}{d x} (u1 \cdot v2) = \frac{d}{d x} (u1) \cdot v2 + u1 \cdot \frac{d}{d x} (v2)

2) Let's differentiate

u1

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

ⓘ The derivative of x^n is n*x^(n-1):

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

2 x

2 x

3) Let's perform the differentiation

v2

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

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

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

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

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

ⓘ Adding 0 changes nothing:

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

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

■The derivative of x^n is n*x^(n-1):

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

- 2 x

- 2 x

4) Let's evaluate the resulting derivative (

u1 \cdot v2

)':

2 x \cdot (1 - x^{2}) + x^{2} \cdot (- 2 x)

ⓘ Multiplying powers with the same base adds the exponents:

x^{2} x

x^{3}

2 x \cdot (1 - x^{2}) - 2 x^{3}

ⓘ Expand by multiplying each term inside the parentheses:

2 x \cdot (1 - x^{2})

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

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

ⓘ The minus sign is placed in front of the product:

2 \cdot (- x^{3})

- 2 x^{3}

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

ⓘ Like terms are grouped by adding their coefficients:

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

-4 x^{3}

2 x - 4 x^{3}

5) Then you can resume the differentiation:

ⅰ

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

2 x - 4 x^{3}

▶Derivative:

2 x - 4 x^{3}