derivative of (x^3-2x+4/(x^3)) — step-by-step solution

To simplify differentiation, we rewrite

\frac{4}{x^{3}}

4 x^{-3}

Let's perform the differentiation:

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

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

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

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

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

We differentiate using:

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

(where f =

x

)

ⓘ The derivative of x^n is n*x^(n-1):

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

3 x^{2}

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

-3 x^{-4}

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

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

-2

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

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

3 x^{2} - 2 + 4 \cdot (-3 x^{-4})

ⓘ Multiplications are done before additions and subtractions. A single minus sign does not cancel out, the product of a negative number by a positive number is negative:

4 \cdot (-3)

-12

3 x^{2} - 2 - 12 x^{-4}

ⓘ The factor is moved to the numerator or the denominator by changing the sign of its exponent, for a nonzero base:

12 x^{-4}

\frac{12}{x^{4}}

3 x^{2} - 2 - \frac{12}{x^{4}}

▶Derivative:

3 x^{2} - \frac{12}{x^{4}} - 2