Calculus Examples

$f (x) = 5 x^{2} + 10 x + 3$

Find the first derivative of the function.

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By the Sum Rule, the derivative of $5 x^{2} + 10 x + 3$ with respect to $x$ is $\frac{d}{d x} [5 x^{2}] + \frac{d}{d x} [10 x] + \frac{d}{d x} [3]$ .

$\frac{d}{d x} [5 x^{2}] + \frac{d}{d x} [10 x] + \frac{d}{d x} [3]$

Evaluate $\frac{d}{d x} [5 x^{2}]$ .

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Since $5$ is constant with respect to $x$ , the derivative of $5 x^{2}$ with respect to $x$ is $5 \frac{d}{d x} [x^{2}]$ .

$5 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [10 x] + \frac{d}{d x} [3]$

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

$5 (2 x) + \frac{d}{d x} [10 x] + \frac{d}{d x} [3]$

Multiply $2$ by $5$ .

$10 x + \frac{d}{d x} [10 x] + \frac{d}{d x} [3]$

Evaluate $\frac{d}{d x} [10 x]$ .

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Since $10$ is constant with respect to $x$ , the derivative of $10 x$ with respect to $x$ is $10 \frac{d}{d x} [x]$ .

$10 x + 10 \frac{d}{d x} [x] + \frac{d}{d x} [3]$

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

$10 x + 10 \cdot 1 + \frac{d}{d x} [3]$

Multiply $10$ by $1$ .

$10 x + 10 + \frac{d}{d x} [3]$

Differentiate using the Constant Rule.

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Since $3$ is constant with respect to $x$ , the derivative of $3$ with respect to $x$ is $0$ .

$10 x + 10 + 0$

Add $10 x + 10$ and $0$ .

$10 x + 10$

Find the second derivative of the function.

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By the Sum Rule, the derivative of $10 x + 10$ with respect to $x$ is $\frac{d}{d x} [10 x] + \frac{d}{d x} [10]$ .

$f'' (x) = \frac{d}{d x} (10 x) + \frac{d}{d x} (10)$

Evaluate $\frac{d}{d x} [10 x]$ .

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Since $10$ is constant with respect to $x$ , the derivative of $10 x$ with respect to $x$ is $10 \frac{d}{d x} [x]$ .

$f'' (x) = 10 \frac{d}{d x} (x) + \frac{d}{d x} (10)$

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

$f'' (x) = 10 \cdot 1 + \frac{d}{d x} (10)$

Multiply $10$ by $1$ .

$f'' (x) = 10 + \frac{d}{d x} (10)$

Differentiate using the Constant Rule.

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Since $10$ is constant with respect to $x$ , the derivative of $10$ with respect to $x$ is $0$ .

$f'' (x) = 10 + 0$

Add $10$ and $0$ .

$f'' (x) = 10$

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

$10 x + 10 = 0$

Subtract $10$ from both sides of the equation.

$10 x = - 10$

Divide each term by $10$ and simplify.

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Divide each term in $10 x = - 10$ by $10$ .

$\frac{10 x}{10} = \frac{- 10}{10}$

Reduce the expression by cancelling the common factors.

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Cancel the common factor.

$\frac{10 x}{10} = \frac{- 10}{10}$

Divide $x$ by $1$ .

$x = \frac{- 10}{10}$

Divide $- 10$ by $10$ .

$x = - 1$

Evaluate the second derivative at $x = - 1$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$10$

$x = - 1$ is a local minimum because the value of the second derivative is positive. This is referred to as the second derivative test.

$x = - 1$ is a local minimum

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