derivative of (x-2/x) — step-by-step solution

To ease differentiation, the expression

- \frac{2}{x}

is turned into

-2 x^{-1}

Let's differentiate:

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

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

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

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

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

We differentiate using:

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

(where f =

x

)

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

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

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

■The derivative of x^n is n*x^(n-1):

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

-1 \cdot x^{-2}

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

ⓘ Multiplications are done before additions and subtractions. Two minus signs cancel out, the product of two negative numbers is positive:

-2 \cdot (-1)

2

1 + 2 x^{-2}

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

2 x^{-2}

\frac{2}{x^{2}}

1 + \frac{2}{x^{2}}

▶Derivative:

\frac{2}{x^{2}} + 1