Calculus Examples

$3 a x^{2} + 2 b x + c$

Step 1

Find the first derivative.

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Step 1.1

Find the first derivative.

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Step 1.1.1

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

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

Step 1.1.2

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

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Step 1.1.2.1

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

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

Step 1.1.2.2

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

$3 x^{2} \cdot 1 + \frac{d}{d a} [2 b x] + \frac{d}{d a} [c]$

Step 1.1.2.3

Multiply $3$ by $1$ .

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

Step 1.1.3

Differentiate using the Constant Rule.

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Step 1.1.3.1

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

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

Step 1.1.3.2

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

$3 x^{2} + 0 + 0$

Step 1.1.4

Combine terms.

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Step 1.1.4.1

Add $3 x^{2}$ and $0$ .

$3 x^{2} + 0$

Step 1.1.4.2

Add $3 x^{2}$ and $0$ .

$f' (a) = 3 x^{2}$

Step 1.2

The first derivative of $f (a)$ with respect to $a$ is $3 x^{2}$ .

$3 x^{2}$

Step 2

Set the first derivative equal to $0$ then solve the equation $3 x^{2} = 0$ .

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Step 2.1

Set the first derivative equal to $0$ .

$3 x^{2} = 0$

Step 2.2

Divide each term in $3 x^{2} = 0$ by $3$ and simplify.

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Step 2.2.1

Divide each term in $3 x^{2} = 0$ by $3$ .

$\frac{3 x^{2}}{3} = \frac{0}{3}$

Step 2.2.2

Simplify the left side.

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Step 2.2.2.1

Cancel the common factor of $3$ .

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Step 2.2.2.1.1

Cancel the common factor.

$\frac{3 x^{2}}{3} = \frac{0}{3}$

Step 2.2.2.1.2

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

$x^{2} = \frac{0}{3}$

Step 2.2.3

Simplify the right side.

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Step 2.2.3.1

Divide $0$ by $3$ .

$x^{2} = 0$

Step 2.3

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{0}$

Step 2.4

Simplify $\pm \sqrt{0}$ .

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Step 2.4.1

Rewrite $0$ as $0^{2}$ .

$x = \pm \sqrt{0^{2}}$

Step 2.4.2

Pull terms out from under the radical, assuming positive real numbers.

$x = \pm 0$

Step 2.4.3

Plus or minus $0$ is $0$ .

$x = 0$

Step 3

Evaluate $3 a x^{2} + 2 b x + c$ at each $a$ value where the derivative is $0$ or undefined.

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Step 3.1

Evaluate at $a = 0$ .

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Step 3.1.1

Substitute $0$ for $a$ .

$3 (0) \cdot x^{2} + 2 b x + c$

Step 3.1.2

Simplify.

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Step 3.1.2.1

Simplify each term.

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Step 3.1.2.1.1

Multiply $3$ by $0$ .

$0 \cdot x^{2} + 2 b x + c$

Step 3.1.2.1.2

Multiply $0$ by $x^{2}$ .

$0 + 2 b x + c$

Step 3.1.2.2

Add $0$ and $2 b x$ .

$2 b x + c$

Step 3.2

List all of the points.

$(0, 2 b x + c)$

Step 4