Calculus Examples

$f (x) = a x^{2} + b x + c$

Step 1

Find the first derivative.

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Step 1.1

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 1.2

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

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Step 1.2.1

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 1.2.2

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 1.2.3

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

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

Step 1.3

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

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Step 1.3.1

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 1.3.2

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 1.3.3

Multiply $b$ by $1$ .

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

Step 1.4

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

$2 a x + b + 0$

Step 1.5

Simplify.

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Step 1.5.1

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

$2 a x + b$

Step 1.5.2

Reorder terms.

$f' (x) = b + 2 a x$

Step 2

Find the second derivative.

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Step 2.1

Differentiate.

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Step 2.1.1

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

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

Step 2.1.2

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

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

Step 2.2

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

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Step 2.2.1

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

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

Step 2.2.2

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

$0 + 2 a \cdot 1$

Step 2.2.3

Multiply $2$ by $1$ .

$0 + 2 a$

Step 2.3

Add $0$ and $2 a$ .

$f'' (x) = 2 a$

Step 3

Since $2 a$ is constant with respect to $x$ , the derivative of $2 a$ with respect to $x$ is $0$ .

$f''' (x) = 0$

Step 4

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

$f^{4} (x) = 0$

Step 5

The fourth derivative of $f (x)$ with respect to $x$ is $0$ .

$0$