double derivative of 7/(x-7) — step-by-step solution

To simplify differentiation, we rewrite

\frac{7}{x - 7}

7 \cdot {(x - 7)}^{-1}

1) Let's differentiate:

\frac{d}{d x} (7 \cdot {(x - 7)}^{-1})

We differentiate using:

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

(where f =

x - 7

)

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

\frac{d}{d x} (7 \cdot {(x - 7)}^{-1})

7 \cdot \frac{d}{d x} ({(x - 7)}^{-1})

■The derivative of x^n is n*x^(n-1):

\frac{d}{d x} ({(x - 7)}^{-1})

-1 \cdot \frac{d}{d x} (x - 7) \cdot {(x - 7)}^{-2}

7 \cdot (-1 \cdot \frac{d}{d x} (x - 7) \cdot {(x - 7)}^{-2})

ⓘ A single minus sign does not cancel out, the product of a negative number by a positive number is negative:

7 \cdot (-1)

-7

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

\frac{d}{d x} (x - 7)

1 + 0

-7 \cdot (1 + 0) \cdot {(x - 7)}^{-2}

ⓘ Adding 0 changes nothing:

1 + 0

1

-7 \cdot {(x - 7)}^{-2}

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

- 7 \cdot {(x - 7)}^{-2}

- \frac{7}{{(x - 7)}^{2}}

- \frac{7}{{(x - 7)}^{2}}

To ease differentiation, the expression

- \frac{7}{{(x - 7)}^{2}}

is turned into

-7 \cdot {(x - 7)}^{-2}

2) Let's perform the differentiation:

\frac{d}{d x} (-7 \cdot {(x - 7)}^{-2})