Algebra Examples

$\frac{k}{x^{n}}$

Step 1

Differentiate using the Constant Multiple Rule.

Tap for more steps...

Step 1.1

Since $k$ is constant with respect to $n$ , the derivative of $\frac{k}{x^{n}}$ with respect to $n$ is $k \frac{d}{d n} [\frac{1}{x^{n}}]$ .

$k \frac{d}{d n} [\frac{1}{x^{n}}]$

Step 1.2

Apply basic rules of exponents.

Tap for more steps...

Step 1.2.1

Rewrite $\frac{1}{x^{n}}$ as ${(x^{n})}^{- 1}$ .

$k \frac{d}{d n} [{(x^{n})}^{- 1}]$

Step 1.2.2

Multiply the exponents in ${(x^{n})}^{- 1}$ .

Tap for more steps...

Step 1.2.2.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$k \frac{d}{d n} [x^{n \cdot - 1}]$

Step 1.2.2.2

Move $- 1$ to the left of $n$ .

$k \frac{d}{d n} [x^{- 1 \cdot n}]$

Step 1.2.2.3

Rewrite $- 1 n$ as $- n$ .

$k \frac{d}{d n} [x^{- n}]$

Step 2

Differentiate using the Generalized Power Rule which states that $\frac{d}{d n} [f^{g}]$ is $f^{g} (f' \frac{g}{f} + g' \ln (f))$ where $f = x$ and $g = - n$ .

$k (\frac{d}{d n} [x] (- n x^{- n - 1}) + \frac{d}{d n} [- n] (x^{- n} \ln (x)))$

Step 3

Differentiate.

Tap for more steps...

Step 3.1

Since $x$ is constant with respect to $n$ , the derivative of $x$ with respect to $n$ is $0$ .

$k (0 (- n x^{- n - 1}) + \frac{d}{d n} [- n] (x^{- n} \ln (x)))$

Step 3.2

Simplify the expression.

Tap for more steps...

Step 3.2.1

Multiply $- 1$ by $0$ .

$k (0 (n x^{- n - 1}) + \frac{d}{d n} [- n] (x^{- n} \ln (x)))$

Step 3.2.2

Multiply $n$ by $0$ .

$k (0 x^{- n - 1} + \frac{d}{d n} [- n] (x^{- n} \ln (x)))$

Step 3.2.3

Multiply $0$ by $x^{- n - 1}$ .

$k (0 + \frac{d}{d n} [- n] (x^{- n} \ln (x)))$

Step 3.2.4

Add $0$ and $\frac{d}{d n} [- n] (x^{- n} \ln (x))$ .

$k (\frac{d}{d n} [- n] (x^{- n} \ln (x)))$

Step 3.3

Since $- 1$ is constant with respect to $n$ , the derivative of $- n$ with respect to $n$ is $- \frac{d}{d n} [n]$ .

$k (- \frac{d}{d n} [n]) (x^{- n} \ln (x))$

Step 3.4

Differentiate using the Power Rule which states that $\frac{d}{d n} [n^{n}]$ is $n \cdot n^{n - 1}$ where $n = 1$ .

$k (- 1 \cdot 1) (x^{- n} \ln (x))$

Step 3.5

Simplify the expression.

Tap for more steps...

Step 3.5.1

Multiply $- 1$ by $1$ .

$k \cdot - 1 (x^{- n} \ln (x))$

Step 3.5.2

Move $- 1$ to the left of $k$ .

$- 1 \cdot k (x^{- n} \ln (x))$

Step 3.5.3

Rewrite $- 1 k$ as $- k$ .

$- k (x^{- n} \ln (x))$

Step 4

Simplify.

Tap for more steps...

Step 4.1

Reorder the factors of $- k x^{- n} \ln (x)$ .

$- x^{- n} k \ln (x)$

Step 4.2

Reorder factors in $- x^{- n} k \ln (x)$ .

$- k x^{- n} \ln (x)$