Calculus Examples

${(x^{2} - y^{2})}^{3} = 3 a^{4} x^{2}$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d a} ({(x^{2} - y^{2})}^{3}) = \frac{d}{d a} (3 a^{4} x^{2})$

Step 2

Differentiate the left side of the equation.

Tap for more steps...

Step 2.1

Use the Binomial Theorem.

$\frac{d}{d a} [{(x^{2})}^{3} + 3 {(x^{2})}^{2} (- y^{2}) + 3 x^{2} {(- y^{2})}^{2} + {(- y^{2})}^{3}]$

Step 2.2

Differentiate.

Tap for more steps...

Step 2.2.1

Simplify each term.

Tap for more steps...

Step 2.2.1.1

Multiply the exponents in ${(x^{2})}^{3}$ .

Tap for more steps...

Step 2.2.1.1.1

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

$\frac{d}{d a} [x^{2 \cdot 3} + 3 {(x^{2})}^{2} (- y^{2}) + 3 x^{2} {(- y^{2})}^{2} + {(- y^{2})}^{3}]$

Step 2.2.1.1.2

Multiply $2$ by $3$ .

$\frac{d}{d a} [x^{6} + 3 {(x^{2})}^{2} (- y^{2}) + 3 x^{2} {(- y^{2})}^{2} + {(- y^{2})}^{3}]$

Step 2.2.1.2

Rewrite using the commutative property of multiplication.

$\frac{d}{d a} [x^{6} + 3 \cdot - 1 {(x^{2})}^{2} y^{2} + 3 x^{2} {(- y^{2})}^{2} + {(- y^{2})}^{3}]$

Step 2.2.1.3

Multiply $3$ by $- 1$ .

$\frac{d}{d a} [x^{6} - 3 {(x^{2})}^{2} y^{2} + 3 x^{2} {(- y^{2})}^{2} + {(- y^{2})}^{3}]$

Step 2.2.1.4

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

Tap for more steps...

Step 2.2.1.4.1

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

$\frac{d}{d a} [x^{6} - 3 x^{2 \cdot 2} y^{2} + 3 x^{2} {(- y^{2})}^{2} + {(- y^{2})}^{3}]$

Step 2.2.1.4.2

Multiply $2$ by $2$ .

$\frac{d}{d a} [x^{6} - 3 x^{4} y^{2} + 3 x^{2} {(- y^{2})}^{2} + {(- y^{2})}^{3}]$

Step 2.2.1.5

Apply the product rule to $- y^{2}$ .

$\frac{d}{d a} [x^{6} - 3 x^{4} y^{2} + 3 x^{2} ({(- 1)}^{2} {(y^{2})}^{2}) + {(- y^{2})}^{3}]$

Step 2.2.1.6

Rewrite using the commutative property of multiplication.

$\frac{d}{d a} [x^{6} - 3 x^{4} y^{2} + 3 \cdot {(- 1)}^{2} x^{2} {(y^{2})}^{2} + {(- y^{2})}^{3}]$

Step 2.2.1.7

Raise $- 1$ to the power of $2$ .

$\frac{d}{d a} [x^{6} - 3 x^{4} y^{2} + 3 \cdot 1 x^{2} {(y^{2})}^{2} + {(- y^{2})}^{3}]$

Step 2.2.1.8

Multiply $3$ by $1$ .

$\frac{d}{d a} [x^{6} - 3 x^{4} y^{2} + 3 x^{2} {(y^{2})}^{2} + {(- y^{2})}^{3}]$

Step 2.2.1.9

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

Tap for more steps...

Step 2.2.1.9.1

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

$\frac{d}{d a} [x^{6} - 3 x^{4} y^{2} + 3 x^{2} y^{2 \cdot 2} + {(- y^{2})}^{3}]$

Step 2.2.1.9.2

Multiply $2$ by $2$ .

$\frac{d}{d a} [x^{6} - 3 x^{4} y^{2} + 3 x^{2} y^{4} + {(- y^{2})}^{3}]$

Step 2.2.1.10

Apply the product rule to $- y^{2}$ .

$\frac{d}{d a} [x^{6} - 3 x^{4} y^{2} + 3 x^{2} y^{4} + {(- 1)}^{3} {(y^{2})}^{3}]$

Step 2.2.1.11

Raise $- 1$ to the power of $3$ .

$\frac{d}{d a} [x^{6} - 3 x^{4} y^{2} + 3 x^{2} y^{4} - {(y^{2})}^{3}]$

Step 2.2.1.12

Multiply the exponents in ${(y^{2})}^{3}$ .

Tap for more steps...

Step 2.2.1.12.1

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

$\frac{d}{d a} [x^{6} - 3 x^{4} y^{2} + 3 x^{2} y^{4} - y^{2 \cdot 3}]$

Step 2.2.1.12.2

Multiply $2$ by $3$ .

$\frac{d}{d a} [x^{6} - 3 x^{4} y^{2} + 3 x^{2} y^{4} - y^{6}]$

Step 2.2.2

By the Sum Rule, the derivative of $x^{6} - 3 x^{4} y^{2} + 3 x^{2} y^{4} - y^{6}$ with respect to $a$ is $\frac{d}{d a} [x^{6}] + \frac{d}{d a} [- 3 x^{4} y^{2}] + \frac{d}{d a} [3 x^{2} y^{4}] + \frac{d}{d a} [- y^{6}]$ .

$\frac{d}{d a} [x^{6}] + \frac{d}{d a} [- 3 x^{4} y^{2}] + \frac{d}{d a} [3 x^{2} y^{4}] + \frac{d}{d a} [- y^{6}]$

Step 2.2.3

Since $x^{6}$ is constant with respect to $a$ , the derivative of $x^{6}$ with respect to $a$ is $0$ .

$0 + \frac{d}{d a} [- 3 x^{4} y^{2}] + \frac{d}{d a} [3 x^{2} y^{4}] + \frac{d}{d a} [- y^{6}]$

Step 2.2.4

Add $0$ and $\frac{d}{d a} [- 3 x^{4} y^{2}]$ .

$\frac{d}{d a} [- 3 x^{4} y^{2}] + \frac{d}{d a} [3 x^{2} y^{4}] + \frac{d}{d a} [- y^{6}]$

Step 2.2.5

Since $- 3 x^{4}$ is constant with respect to $a$ , the derivative of $- 3 x^{4} y^{2}$ with respect to $a$ is $- 3 x^{4} \frac{d}{d a} [y^{2}]$ .

$- 3 x^{4} \frac{d}{d a} [y^{2}] + \frac{d}{d a} [3 x^{2} y^{4}] + \frac{d}{d a} [- y^{6}]$

Step 2.3

Differentiate using the chain rule, which states that $\frac{d}{d a} [f (g (a))]$ is $f' (g (a)) g' (a)$ where $f (a) = a^{2}$ and $g (a) = y$ .

Tap for more steps...

Step 2.3.1

To apply the Chain Rule, set $u_{1}$ as $y$ .

$- 3 x^{4} (\frac{d}{d u_{1}} [{u_{1}}^{2}] \frac{d}{d a} [y]) + \frac{d}{d a} [3 x^{2} y^{4}] + \frac{d}{d a} [- y^{6}]$

Step 2.3.2

Differentiate using the Power Rule which states that $\frac{d}{d u_{1}} [{u_{1}}^{n}]$ is $n {u_{1}}^{n - 1}$ where $n = 2$ .

$- 3 x^{4} (2 u_{1} \frac{d}{d a} [y]) + \frac{d}{d a} [3 x^{2} y^{4}] + \frac{d}{d a} [- y^{6}]$

Step 2.3.3

Replace all occurrences of $u_{1}$ with $y$ .

$- 3 x^{4} (2 y \frac{d}{d a} [y]) + \frac{d}{d a} [3 x^{2} y^{4}] + \frac{d}{d a} [- y^{6}]$

Step 2.4

Multiply $2$ by $- 3$ .

$- 6 x^{4} (y \frac{d}{d a} [y]) + \frac{d}{d a} [3 x^{2} y^{4}] + \frac{d}{d a} [- y^{6}]$

Step 2.5

Rewrite $\frac{d}{d a} [y]$ as $y'$ .

$- 6 x^{4} y y' + \frac{d}{d a} [3 x^{2} y^{4}] + \frac{d}{d a} [- y^{6}]$

Step 2.6

Since $3 x^{2}$ is constant with respect to $a$ , the derivative of $3 x^{2} y^{4}$ with respect to $a$ is $3 x^{2} \frac{d}{d a} [y^{4}]$ .

$- 6 x^{4} y y' + 3 x^{2} \frac{d}{d a} [y^{4}] + \frac{d}{d a} [- y^{6}]$

Step 2.7

Differentiate using the chain rule, which states that $\frac{d}{d a} [f (g (a))]$ is $f' (g (a)) g' (a)$ where $f (a) = a^{4}$ and $g (a) = y$ .

Tap for more steps...

Step 2.7.1

To apply the Chain Rule, set $u_{2}$ as $y$ .

$- 6 x^{4} y y' + 3 x^{2} (\frac{d}{d u_{2}} [{u_{2}}^{4}] \frac{d}{d a} [y]) + \frac{d}{d a} [- y^{6}]$

Step 2.7.2

Differentiate using the Power Rule which states that $\frac{d}{d u_{2}} [{u_{2}}^{n}]$ is $n {u_{2}}^{n - 1}$ where $n = 4$ .

$- 6 x^{4} y y' + 3 x^{2} (4 {u_{2}}^{3} \frac{d}{d a} [y]) + \frac{d}{d a} [- y^{6}]$

Step 2.7.3

Replace all occurrences of $u_{2}$ with $y$ .

$- 6 x^{4} y y' + 3 x^{2} (4 y^{3} \frac{d}{d a} [y]) + \frac{d}{d a} [- y^{6}]$

Step 2.8

Multiply $4$ by $3$ .

$- 6 x^{4} y y' + 12 x^{2} (y^{3} \frac{d}{d a} [y]) + \frac{d}{d a} [- y^{6}]$

Step 2.9

Rewrite $\frac{d}{d a} [y]$ as $y'$ .

$- 6 x^{4} y y' + 12 x^{2} y^{3} y' + \frac{d}{d a} [- y^{6}]$

Step 2.10

Since $- 1$ is constant with respect to $a$ , the derivative of $- y^{6}$ with respect to $a$ is $- \frac{d}{d a} [y^{6}]$ .

$- 6 x^{4} y y' + 12 x^{2} y^{3} y' - \frac{d}{d a} [y^{6}]$

Step 2.11

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

Tap for more steps...

Step 2.11.1

To apply the Chain Rule, set $u_{3}$ as $y$ .

$- 6 x^{4} y y' + 12 x^{2} y^{3} y' - (\frac{d}{d u_{3}} [{u_{3}}^{6}] \frac{d}{d a} [y])$

Step 2.11.2

Differentiate using the Power Rule which states that $\frac{d}{d u_{3}} [{u_{3}}^{n}]$ is $n {u_{3}}^{n - 1}$ where $n = 6$ .

$- 6 x^{4} y y' + 12 x^{2} y^{3} y' - (6 {u_{3}}^{5} \frac{d}{d a} [y])$

Step 2.11.3

Replace all occurrences of $u_{3}$ with $y$ .

$- 6 x^{4} y y' + 12 x^{2} y^{3} y' - (6 y^{5} \frac{d}{d a} [y])$

Step 2.12

Multiply $6$ by $- 1$ .

$- 6 x^{4} y y' + 12 x^{2} y^{3} y' - 6 (y^{5} \frac{d}{d a} [y])$

Step 2.13

Rewrite $\frac{d}{d a} [y]$ as $y'$ .

$- 6 x^{4} y y' + 12 x^{2} y^{3} y' - 6 y^{5} y'$

Step 2.14

Reorder terms.

$- 6 x^{4} y y' + 12 y^{3} x^{2} y' - 6 y^{5} y'$

Step 3

Differentiate the right side of the equation.

Tap for more steps...

Step 3.1

Since $3 x^{2}$ is constant with respect to $a$ , the derivative of $3 a^{4} x^{2}$ with respect to $a$ is $3 x^{2} \frac{d}{d a} [a^{4}]$ .

$3 x^{2} \frac{d}{d a} [a^{4}]$

Step 3.2

Differentiate using the Power Rule which states that $\frac{d}{d a} [a^{n}]$ is $n a^{n - 1}$ where $n = 4$ .

$3 x^{2} (4 a^{3})$

Step 3.3

Simplify the expression.

Tap for more steps...

Step 3.3.1

Multiply $4$ by $3$ .

$12 x^{2} a^{3}$

Step 3.3.2

Reorder the factors of $12 x^{2} a^{3}$ .

$12 a^{3} x^{2}$

Step 4

Reform the equation by setting the left side equal to the right side.

$- 6 x^{4} y y' + 12 y^{3} x^{2} y' - 6 y^{5} y' = 12 a^{3} x^{2}$

Step 5

Replace $y'$ with $\frac{d y}{d a}$ .

$\frac{d y}{d a} = 12 a^{3} x^{2}$