Find the Derivative Using Quotient Rule - d/dx k/(x^n)

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

Step 1

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

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

Step 2

Differentiate.

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

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

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

$\frac{x^{n} \cdot 0 - k (n x^{n - 1})}{{(x^{n})}^{2}}$

Step 3

Simplify.

Simplify the numerator.

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

$\frac{0 - k n x^{n - 1}}{{(x^{n})}^{2}}$

Subtract $k n x^{n - 1}$ from $0$ .

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

Combine terms.

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

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

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

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

$\frac{- k n x^{n - 1}}{x^{2 n}}$

Cancel the common factor of $x^{n - 1}$ and $x^{2 n}$ .

Factor $x^{2 n}$ out of $- k n x^{n - 1}$ .

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

Cancel the common factors.

Multiply by $1$ .

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

Cancel the common factor.

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

Rewrite the expression.

$\frac{- k n x^{- n - 1}}{1}$

Divide $- k n x^{- n - 1}$ by $1$ .

$- k n x^{- n - 1}$

Reorder the factors of $- k n x^{- n - 1}$ .

$- x^{- n - 1} k n$

Reorder factors in $- x^{- n - 1} k n$ .

$- k n x^{- n - 1}$

Find the Derivative Using Quotient Rule - d/dx k/(x^n)

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