Algebra Examples

$f {(y e s)}^{- 1}$

Step 1

Since $f$ is constant with respect to $y$ , the derivative of $f {(y e s)}^{- 1}$ with respect to $y$ is $f \frac{d}{d y} [{(y e s)}^{- 1}]$ .

$f \frac{d}{d y} [{(y e s)}^{- 1}]$

Step 2

Differentiate using the chain rule, which states that $\frac{d}{d y} [f (g (y))]$ is $f' (g (y)) g' (y)$ where $f (y) = y^{- 1}$ and $g (y) = y e s$ .

Tap for more steps...

Step 2.1

To apply the Chain Rule, set $u$ as $y e s$ .

$f (\frac{d}{d u} [u^{- 1}] \frac{d}{d y} [y e s])$

Step 2.2

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

$f (- u^{- 2} \frac{d}{d y} [y e s])$

Step 2.3

Replace all occurrences of $u$ with $y e s$ .

$f (- {(y e s)}^{- 2} \frac{d}{d y} [y e s])$

Step 3

Differentiate.

Tap for more steps...

Step 3.1

Since $e s$ is constant with respect to $y$ , the derivative of $y e s$ with respect to $y$ is $e s \frac{d}{d y} [y]$ .

$f (- {(y e s)}^{- 2} (e s \frac{d}{d y} [y]))$

Step 3.2

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

$f (- {(y e s)}^{- 2} (e s \cdot 1))$

Step 3.3

Multiply $e$ by $1$ .

$f (- {(y e s)}^{- 2} (s \cdot e))$

Step 4

Simplify.

Tap for more steps...

Step 4.1

Apply the product rule to $y e s$ .

$f (- ({(y e)}^{- 2} s^{- 2}) (s e))$

Step 4.2

Apply the product rule to $y e$ .

$f (- (y^{- 2} e^{- 2} s^{- 2}) (s e))$

Step 4.3

Combine terms.

Tap for more steps...

Step 4.3.1

Raise $e$ to the power of $1$ .

$f (- y^{- 2} s^{- 2} (s (e^{- 2} e^{1})))$

Step 4.3.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f (- y^{- 2} s^{- 2} (s e^{- 2 + 1}))$

Step 4.3.3

Add $- 2$ and $1$ .

$f (- y^{- 2} s^{- 2} (s e^{- 1}))$

Step 4.3.4

Raise $s$ to the power of $1$ .

$f (- y^{- 2} (s^{1} s^{- 2}) e^{- 1})$

Step 4.3.5

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f (- y^{- 2} s^{1 - 2} e^{- 1})$

Step 4.3.6

Subtract $2$ from $1$ .

$f (- y^{- 2} s^{- 1} e^{- 1})$

Step 4.4

Reorder the factors of $f (- y^{- 2} s^{- 1} e^{- 1})$ .

$- e^{- 1} f s^{- 1} y^{- 2}$

Step 4.5

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$- \frac{1}{e} f s^{- 1} y^{- 2}$

Step 4.6

Combine $f$ and $\frac{1}{e}$ .

$- \frac{f}{e} s^{- 1} y^{- 2}$

Step 4.7

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$- \frac{f}{e} \cdot \frac{1}{s} y^{- 2}$

Step 4.8

Multiply $\frac{1}{s}$ by $\frac{f}{e}$ .

$- \frac{f}{s e} y^{- 2}$

Step 4.9

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$- \frac{f}{s e} \cdot \frac{1}{y^{2}}$

Step 4.10

Multiply $\frac{1}{y^{2}}$ by $\frac{f}{s e}$ .

$- \frac{f}{y^{2} s e}$