Algebra Examples

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

Step 1

Differentiate using the Quotient Rule which states that $\frac{d}{d n} [\frac{f (n)}{g (n)}]$ is $\frac{g (n) \frac{d}{d n} [f (n)] - f (n) \frac{d}{d n} [g (n)]}{{g (n)}^{2}}$ where $f (n) = k$ and $g (n) = x^{n}$ .

$\frac{x^{n} \frac{d}{d n} [k] - k \frac{d}{d n} [x^{n}]}{{(x^{n})}^{2}}$

Step 2

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

$\frac{x^{n} \cdot 0 - k \frac{d}{d n} [x^{n}]}{{(x^{n})}^{2}}$

Step 3

Differentiate using the Exponential Rule which states that $\frac{d}{d n} [a^{n}]$ is $a^{n} \ln (a)$ where $a$ = $x$ .

$\frac{x^{n} \cdot 0 - k (x^{n} \ln (x))}{{(x^{n})}^{2}}$

Step 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

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

$\frac{0 - k x^{n} \ln (x)}{{(x^{n})}^{2}}$

Step 4.1.2

Subtract $k x^{n} \ln (x)$ from $0$ .

$\frac{- k x^{n} \ln (x)}{{(x^{n})}^{2}}$

Step 4.2

Combine terms.

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Step 4.2.1

Multiply the exponents in ${(x^{n})}^{2}$ .

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Step 4.2.1.1

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

$\frac{- k x^{n} \ln (x)}{x^{n \cdot 2}}$

Step 4.2.1.2

Move $2$ to the left of $n$ .

$\frac{- k x^{n} \ln (x)}{x^{2 n}}$

Step 4.2.2

Cancel the common factor of $x^{n}$ and $x^{2 n}$ .

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Step 4.2.2.1

Factor $x^{2 n}$ out of $- k x^{n} \ln (x)$ .

$\frac{x^{2 n} (- k x^{- n} \ln (x))}{x^{2 n}}$

Step 4.2.2.2

Cancel the common factors.

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Step 4.2.2.2.1

Multiply by $1$ .

$\frac{x^{2 n} (- k x^{- n} \ln (x))}{x^{2 n} \cdot 1}$

Step 4.2.2.2.2

Cancel the common factor.

$\frac{x^{2 n} (- k x^{- n} \ln (x))}{x^{2 n} \cdot 1}$

Step 4.2.2.2.3

Rewrite the expression.

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

Step 4.2.2.2.4

Divide $- k x^{- n} \ln (x)$ by $1$ .

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

Step 4.3

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

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

Step 4.4

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

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