derivative of 5-4/x — step-by-step solution

To ease differentiation, the expression

- \frac{4}{x}

is turned into

-4 x^{-1}

Let's perform the differentiation:

\frac{d}{d x} (5 - 4 x^{-1})

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

\frac{d}{d x} (5 - 4 x^{-1})

0 + \frac{d}{d x} (- 4 x^{-1})

0 + \frac{d}{d x} (- 4 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

)

ⓘ Adding 0 changes nothing:

0 + \frac{d}{d x} (- 4 x^{-1})

\frac{d}{d x} (- 4 x^{-1})

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

\frac{d}{d x} (- 4 x^{-1})

-4 \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}

-4 \cdot (-1 \cdot x^{-2})

ⓘ Two minus signs cancel out, the product of two negative numbers is positive:

-4 \cdot (-1)

4

4 x^{-2}

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

4 x^{-2}

\frac{4}{x^{2}}

▶Derivative:

\frac{4}{x^{2}}