Calculus Examples

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

Step 1

Find the first derivative of the function.

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Step 1.1

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

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

Step 1.2

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

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Step 1.2.1

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

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

Step 1.2.2

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

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

Step 1.2.3

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

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

Step 1.3

Differentiate using the Constant Rule.

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Step 1.3.1

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

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

Step 1.3.2

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

$x^{2} + 0 + 0$

Step 1.4

Combine terms.

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Step 1.4.1

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

$x^{2} + 0$

Step 1.4.2

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

$x^{2}$

Step 2

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

$f'' (a) = 0$

Step 3

To find the local maximum and minimum values of the function, set the derivative equal to $0$ and solve.

$x^{2} = 0$

Step 4

Find the first derivative.

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Step 4.1

Find the first derivative.

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Step 4.1.1

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

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

Step 4.1.2

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

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Step 4.1.2.1

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

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

Step 4.1.2.2

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

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

Step 4.1.2.3

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

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

Step 4.1.3

Differentiate using the Constant Rule.

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Step 4.1.3.1

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

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

Step 4.1.3.2

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

$x^{2} + 0 + 0$

Step 4.1.4

Combine terms.

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Step 4.1.4.1

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

$x^{2} + 0$

Step 4.1.4.2

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

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

Step 4.2

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

$x^{2}$

Step 5

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

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Step 5.1

Set the first derivative equal to $0$ .

$x^{2} = 0$

Step 5.2

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

$x = \pm \sqrt{0}$

Step 5.3

Simplify $\pm \sqrt{0}$ .

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Step 5.3.1

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

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

Step 5.3.2

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

$x = \pm 0$

Step 5.3.3

Plus or minus $0$ is $0$ .

$x = 0$

Step 6

Critical points to evaluate.

$a = 0$

Step 7

Evaluate the second derivative at $a = 0$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$0$

Step 8

Since the first derivative test failed, there are no local extrema.

No Local Extrema

Step 9