Calculus Examples

$x^{4} + 4 x^{b} y + a x^{2} y^{2} + 4 x y^{3} + y^{4}$

Step 1

Differentiate.

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Step 1.1

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

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

Step 1.2

Since $x^{4}$ is constant with respect to $a$ , the derivative of $x^{4}$ with respect to $a$ is $0$ .

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

Step 1.3

Since $4 x^{b} y$ is constant with respect to $a$ , the derivative of $4 x^{b} y$ with respect to $a$ is $0$ .

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

Step 2

Evaluate $\frac{d}{d a} [a x^{2} y^{2}]$ .

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Step 2.1

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

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

Step 2.2

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

$0 + 0 + x^{2} y^{2} \cdot 1 + \frac{d}{d a} [4 x y^{3}] + \frac{d}{d a} [y^{4}]$

Step 2.3

Multiply $x^{2}$ by $1$ .

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

Step 3

Differentiate using the Constant Rule.

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Step 3.1

Since $4 x y^{3}$ is constant with respect to $a$ , the derivative of $4 x y^{3}$ with respect to $a$ is $0$ .

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

Step 3.2

Since $y^{4}$ is constant with respect to $a$ , the derivative of $y^{4}$ with respect to $a$ is $0$ .

$0 + 0 + x^{2} y^{2} + 0 + 0$

Step 4

Combine terms.

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Step 4.1

Add $0$ and $0$ .

$0 + x^{2} y^{2} + 0 + 0$

Step 4.2

Add $0$ and $x^{2} y^{2}$ .

$x^{2} y^{2} + 0 + 0$

Step 4.3

Add $x^{2} y^{2}$ and $0$ .

$x^{2} y^{2} + 0$

Step 4.4

Add $x^{2} y^{2}$ and $0$ .

$x^{2} y^{2}$