Calculate the derivative of f(x) = x^3 - 4x^2 + x - 7

We save the definition:

f (x) = x^{3}

- 4 x^{2}

+ x

- 7

Let's perform the differentiation:

\frac{d}{d x} (x^{3} - 4 x^{2} + x - 7)

ⓘ Differentiate a sum by differentiating each term, then adding the derivatives:

\frac{d}{d x} (x^{3} - 4 x^{2} + x - 7)

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

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

+ 1

+ 0

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

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

+ 1

+ 0

ⓘ Differentiating

x^{n}

gives

n x^{n - 1}

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

3 x^{2}

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

2 x

■The derivative of a constant multiple keeps that factor: that of

k u

k \cdot u'

\frac{d}{d x} (- 4 x^{2})

-4 \cdot \frac{d}{d x} (x^{2})

■Adding 0 preserves the original value:

1 + 0

1

3 x^{2} - 4 \cdot 2 x + 1

ⓘ Perform multiplications before additions and subtractions:

4 \cdot 2

8

▶Derivative:

3 x^{2} - 8 x + 1