Calculus Examples

$a x^{2} + b x + c$

Step 1

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

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

Step 2

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

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Step 2.1

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

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

Step 2.2

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

$a (2 x) + \frac{d}{d x} [b x] + \frac{d}{d x} [c]$

Step 2.3

Move $2$ to the left of $a$ .

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

Step 3

Evaluate $\frac{d}{d x} [b x]$ .

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Step 3.1

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

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

Step 3.2

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

$2 a x + b \cdot 1 + \frac{d}{d x} [c]$

Step 3.3

Multiply $b$ by $1$ .

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

Step 4

Since $c$ is constant with respect to $x$ , the derivative of $c$ with respect to $x$ is $0$ .

$2 a x + b + 0$

Step 5

Simplify.

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Step 5.1

Add $2 a x + b$ and $0$ .

$2 a x + b$

Step 5.2

Reorder terms.

$b + 2 a x$