Calculus Examples

$h (x) = 7 x^{2} (6 x + 5^{2})$

Step 1

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = 7 x^{2}$ and $g (x) = 6 x + 5^{2}$ .

$7 x^{2} \frac{d}{d x} [6 x + 5^{2}] + (6 x + 5^{2}) \frac{d}{d x} [7 x^{2}]$

Step 2

Differentiate using the Sum Rule.

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Step 2.1

Raise $5$ to the power of $2$ .

$7 x^{2} \frac{d}{d x} [6 x + 25] + (6 x + 5^{2}) \frac{d}{d x} [7 x^{2}]$

Step 2.2

By the Sum Rule, the derivative of $6 x + 25$ with respect to $x$ is $\frac{d}{d x} [6 x] + \frac{d}{d x} [25]$ .

$7 x^{2} (\frac{d}{d x} [6 x] + \frac{d}{d x} [25]) + (6 x + 5^{2}) \frac{d}{d x} [7 x^{2}]$

Step 3

Evaluate $\frac{d}{d x} [6 x]$ .

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Step 3.1

Since $6$ is constant with respect to $x$ , the derivative of $6 x$ with respect to $x$ is $6 \frac{d}{d x} [x]$ .

$7 x^{2} (6 \frac{d}{d x} [x] + \frac{d}{d x} [25]) + (6 x + 5^{2}) \frac{d}{d x} [7 x^{2}]$

Step 3.2

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

$7 x^{2} (6 \cdot 1 + \frac{d}{d x} [25]) + (6 x + 5^{2}) \frac{d}{d x} [7 x^{2}]$

Step 3.3

Multiply $6$ by $1$ .

$7 x^{2} (6 + \frac{d}{d x} [25]) + (6 x + 5^{2}) \frac{d}{d x} [7 x^{2}]$

Step 4

Differentiate.

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Step 4.1

Since $25$ is constant with respect to $x$ , the derivative of $25$ with respect to $x$ is $0$ .

$7 x^{2} (6 + 0) + (6 x + 5^{2}) \frac{d}{d x} [7 x^{2}]$

Step 4.2

Add $6$ and $0$ .

$7 x^{2} \cdot 6 + (6 x + 5^{2}) \frac{d}{d x} [7 x^{2}]$

Step 4.3

Since $7$ is constant with respect to $x$ , the derivative of $7 x^{2}$ with respect to $x$ is $7 \frac{d}{d x} [x^{2}]$ .

$7 x^{2} \cdot 6 + (6 x + 5^{2}) (7 \frac{d}{d x} [x^{2}])$

Step 4.4

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

$7 x^{2} \cdot 6 + (6 x + 5^{2}) (7 (2 x))$

Step 4.5

Multiply $2$ by $7$ .

$7 x^{2} \cdot 6 + (6 x + 5^{2}) (14 x)$

Step 5

Simplify.

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Step 5.1

Apply the distributive property.

$7 x^{2} \cdot 6 + 6 x (14 x) + 5^{2} (14 x)$

Step 5.2

Combine terms.

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Step 5.2.1

Multiply $6$ by $7$ .

$42 x^{2} + 6 x (14 x) + 5^{2} (14 x)$

Step 5.2.2

Multiply $14$ by $6$ .

$42 x^{2} + 84 x \cdot x + 5^{2} (14 x)$

Step 5.2.3

Raise $x$ to the power of $1$ .

$42 x^{2} + 84 (x^{1} x) + 5^{2} (14 x)$

Step 5.2.4

Raise $x$ to the power of $1$ .

$42 x^{2} + 84 (x^{1} x^{1}) + 5^{2} (14 x)$

Step 5.2.5

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$42 x^{2} + 84 x^{1 + 1} + 5^{2} (14 x)$

Step 5.2.6

Add $1$ and $1$ .

$42 x^{2} + 84 x^{2} + 5^{2} (14 x)$

Step 5.2.7

Raise $5$ to the power of $2$ .

$42 x^{2} + 84 x^{2} + 25 (14 x)$

Step 5.2.8

Multiply $14$ by $25$ .

$42 x^{2} + 84 x^{2} + 350 x$

Step 5.2.9

Add $42 x^{2}$ and $84 x^{2}$ .

$126 x^{2} + 350 x$