Calculus Examples

$a^{2} + b^{2} + (a^{2} + b^{2})$

Step 1

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

$\frac{d}{d a} [a^{2}] + \frac{d}{d a} [b^{2}] + \frac{d}{d a} [a^{2}] + \frac{d}{d a} [b^{2}]$

Step 2

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

$2 a + \frac{d}{d a} [b^{2}] + \frac{d}{d a} [a^{2}] + \frac{d}{d a} [b^{2}]$

Step 3

Since $b^{2}$ is constant with respect to $a$ , the derivative of $b^{2}$ with respect to $a$ is $0$ .

$2 a + 0 + \frac{d}{d a} [a^{2}] + \frac{d}{d a} [b^{2}]$

Step 4

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

$2 a + 0 + 2 a + \frac{d}{d a} [b^{2}]$

Step 5

Since $b^{2}$ is constant with respect to $a$ , the derivative of $b^{2}$ with respect to $a$ is $0$ .

$2 a + 0 + 2 a + 0$

Step 6

Combine terms.

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Step 6.1

Add $2 a$ and $0$ .

$2 a + 2 a + 0$

Step 6.2

Add $2 a$ and $2 a$ .

$4 a + 0$

Step 6.3

Add $4 a$ and $0$ .

$4 a$