Calculus Examples

${(\frac{1}{y^{9}})}^{6}$

Step 1

Differentiate using the chain rule, which states that $\frac{d}{d y} [f (g (y))]$ is $f' (g (y)) g' (y)$ where $f (y) = y^{6}$ and $g (y) = \frac{1}{y^{9}}$ .

Tap for more steps...

Step 1.1

To apply the Chain Rule, set $u$ as $\frac{1}{y^{9}}$ .

$\frac{d}{d u} [u^{6}] \frac{d}{d y} [\frac{1}{y^{9}}]$

Step 1.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = 6$ .

$6 u^{5} \frac{d}{d y} [\frac{1}{y^{9}}]$

Step 1.3

Replace all occurrences of $u$ with $\frac{1}{y^{9}}$ .

$6 {(\frac{1}{y^{9}})}^{5} \frac{d}{d y} [\frac{1}{y^{9}}]$

Step 2

Differentiate using the Power Rule.

Tap for more steps...

Step 2.1

Rewrite $\frac{1}{y^{9}}$ as ${(y^{9})}^{- 1}$ .

$6 {(\frac{1}{y^{9}})}^{5} \frac{d}{d y} [{(y^{9})}^{- 1}]$

Step 2.2

Multiply the exponents in ${(y^{9})}^{- 1}$ .

Tap for more steps...

Step 2.2.1

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

$6 {(\frac{1}{y^{9}})}^{5} \frac{d}{d y} [y^{9 \cdot - 1}]$

Step 2.2.2

Multiply $9$ by $- 1$ .

$6 {(\frac{1}{y^{9}})}^{5} \frac{d}{d y} [y^{- 9}]$

Step 2.3

Differentiate using the Power Rule which states that $\frac{d}{d y} [y^{n}]$ is $n y^{n - 1}$ where $n = - 9$ .

$6 {(\frac{1}{y^{9}})}^{5} (- 9 y^{- 10})$

Step 2.4

Multiply $- 9$ by $6$ .

$- 54 {(\frac{1}{y^{9}})}^{5} y^{- 10}$

Step 3

Simplify.

Tap for more steps...

Step 3.1

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

$- 54 {(\frac{1}{y^{9}})}^{5} \frac{1}{y^{10}}$

Step 3.2

Apply the product rule to $\frac{1}{y^{9}}$ .

$- 54 \frac{1^{5}}{{(y^{9})}^{5}} \frac{1}{y^{10}}$

Step 3.3

Combine terms.

Tap for more steps...

Step 3.3.1

One to any power is one.

$- 54 \frac{1}{{(y^{9})}^{5}} \frac{1}{y^{10}}$

Step 3.3.2

Multiply the exponents in ${(y^{9})}^{5}$ .

Tap for more steps...

Step 3.3.2.1

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

$- 54 \frac{1}{y^{9 \cdot 5}} \frac{1}{y^{10}}$

Step 3.3.2.2

Multiply $9$ by $5$ .

$- 54 \frac{1}{y^{45}} \frac{1}{y^{10}}$

Step 3.3.3

Combine $- 54$ and $\frac{1}{y^{45}}$ .

$\frac{- 54}{y^{45}} \cdot \frac{1}{y^{10}}$

Step 3.3.4

Move the negative in front of the fraction.

$- \frac{54}{y^{45}} \cdot \frac{1}{y^{10}}$

Step 3.3.5

Multiply $\frac{1}{y^{10}}$ by $\frac{54}{y^{45}}$ .

$- \frac{54}{y^{10} y^{45}}$

Step 3.3.6

Multiply $y^{10}$ by $y^{45}$ by adding the exponents.

Tap for more steps...

Step 3.3.6.1

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

$- \frac{54}{y^{10 + 45}}$

Step 3.3.6.2

Add $10$ and $45$ .

$- \frac{54}{y^{55}}$