differentiate 1/(x+1) — step-by-step solution

To ease differentiation, the expression

\frac{1}{x + 1}

is turned into

{(x + 1)}^{-1}

Let's differentiate:

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

We differentiate using:

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

(where f =

x + 1

)

ⓘ The derivative of x^n is n*x^(n-1):

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

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

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

ⓘ Multiplying by -1 gives the opposite:

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

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

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

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

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

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

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

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

\frac{d}{d x} (x + 1)

1 + 0

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

ⓘ Adding 0 changes nothing:

1 + 0

1

▶Derivative:

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