Calculus Examples

$f (x) = 6 x^{2} \ln (7 x^{3} + 10 x^{2})$

Step 1

Since $6$ is constant with respect to $x$ , the derivative of $6 x^{2} \ln (7 x^{3} + 10 x^{2})$ with respect to $x$ is $6 \frac{d}{d x} [x^{2} \ln (7 x^{3} + 10 x^{2})]$ .

$6 \frac{d}{d x} [x^{2} \ln (7 x^{3} + 10 x^{2})]$

Step 2

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 $f (x) = x^{2}$ and $g (x) = \ln (7 x^{3} + 10 x^{2})$ .

$6 (x^{2} \frac{d}{d x} [\ln (7 x^{3} + 10 x^{2})] + \ln (7 x^{3} + 10 x^{2}) \frac{d}{d x} [x^{2}])$

Step 3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \ln (x)$ and $g (x) = 7 x^{3} + 10 x^{2}$ .

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Step 3.1

To apply the Chain Rule, set $u$ as $7 x^{3} + 10 x^{2}$ .

$6 (x^{2} (\frac{d}{d u} [\ln (u)] \frac{d}{d x} [7 x^{3} + 10 x^{2}]) + \ln (7 x^{3} + 10 x^{2}) \frac{d}{d x} [x^{2}])$

Step 3.2

The derivative of $\ln (u)$ with respect to $u$ is $\frac{1}{u}$ .

$6 (x^{2} (\frac{1}{u} \frac{d}{d x} [7 x^{3} + 10 x^{2}]) + \ln (7 x^{3} + 10 x^{2}) \frac{d}{d x} [x^{2}])$

Step 3.3

Replace all occurrences of $u$ with $7 x^{3} + 10 x^{2}$ .

$6 (x^{2} (\frac{1}{7 x^{3} + 10 x^{2}} \frac{d}{d x} [7 x^{3} + 10 x^{2}]) + \ln (7 x^{3} + 10 x^{2}) \frac{d}{d x} [x^{2}])$

Step 4

Differentiate.

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Step 4.1

Combine $\frac{1}{7 x^{3} + 10 x^{2}}$ and $x^{2}$ .

$6 (\frac{x^{2}}{7 x^{3} + 10 x^{2}} \frac{d}{d x} [7 x^{3} + 10 x^{2}] + \ln (7 x^{3} + 10 x^{2}) \frac{d}{d x} [x^{2}])$

Step 4.2

By the Sum Rule, the derivative of $7 x^{3} + 10 x^{2}$ with respect to $x$ is $\frac{d}{d x} [7 x^{3}] + \frac{d}{d x} [10 x^{2}]$ .

$6 (\frac{x^{2}}{7 x^{3} + 10 x^{2}} (\frac{d}{d x} [7 x^{3}] + \frac{d}{d x} [10 x^{2}]) + \ln (7 x^{3} + 10 x^{2}) \frac{d}{d x} [x^{2}])$

Step 4.3

Since $7$ is constant with respect to $x$ , the derivative of $7 x^{3}$ with respect to $x$ is $7 \frac{d}{d x} [x^{3}]$ .

$6 (\frac{x^{2}}{7 x^{3} + 10 x^{2}} (7 \frac{d}{d x} [x^{3}] + \frac{d}{d x} [10 x^{2}]) + \ln (7 x^{3} + 10 x^{2}) \frac{d}{d x} [x^{2}])$

Step 4.4

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

$6 (\frac{x^{2}}{7 x^{3} + 10 x^{2}} (7 (3 x^{2}) + \frac{d}{d x} [10 x^{2}]) + \ln (7 x^{3} + 10 x^{2}) \frac{d}{d x} [x^{2}])$

Step 4.5

Multiply $3$ by $7$ .

$6 (\frac{x^{2}}{7 x^{3} + 10 x^{2}} (21 x^{2} + \frac{d}{d x} [10 x^{2}]) + \ln (7 x^{3} + 10 x^{2}) \frac{d}{d x} [x^{2}])$

Step 4.6

Since $10$ is constant with respect to $x$ , the derivative of $10 x^{2}$ with respect to $x$ is $10 \frac{d}{d x} [x^{2}]$ .

$6 (\frac{x^{2}}{7 x^{3} + 10 x^{2}} (21 x^{2} + 10 \frac{d}{d x} [x^{2}]) + \ln (7 x^{3} + 10 x^{2}) \frac{d}{d x} [x^{2}])$

Step 4.7

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

$6 (\frac{x^{2}}{7 x^{3} + 10 x^{2}} (21 x^{2} + 10 (2 x)) + \ln (7 x^{3} + 10 x^{2}) \frac{d}{d x} [x^{2}])$

Step 4.8

Multiply $2$ by $10$ .

$6 (\frac{x^{2}}{7 x^{3} + 10 x^{2}} (21 x^{2} + 20 x) + \ln (7 x^{3} + 10 x^{2}) \frac{d}{d x} [x^{2}])$

Step 4.9

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

$6 (\frac{x^{2}}{7 x^{3} + 10 x^{2}} (21 x^{2} + 20 x) + \ln (7 x^{3} + 10 x^{2}) (2 x))$

Step 5

Simplify.

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Step 5.1

Apply the distributive property.

$6 (\frac{x^{2}}{7 x^{3} + 10 x^{2}} (21 x^{2} + 20 x)) + 6 (\ln (7 x^{3} + 10 x^{2}) (2 x))$

Step 5.2

Combine terms.

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Step 5.2.1

Combine $\frac{x^{2}}{7 x^{3} + 10 x^{2}}$ and $6$ .

$\frac{x^{2} \cdot 6}{7 x^{3} + 10 x^{2}} (21 x^{2} + 20 x) + 6 (\ln (7 x^{3} + 10 x^{2}) (2 x))$

Step 5.2.2

Move $6$ to the left of $x^{2}$ .

$\frac{6 \cdot x^{2}}{7 x^{3} + 10 x^{2}} (21 x^{2} + 20 x) + 6 (\ln (7 x^{3} + 10 x^{2}) (2 x))$

Step 5.2.3

Multiply $2$ by $6$ .

$\frac{6 x^{2}}{7 x^{3} + 10 x^{2}} (21 x^{2} + 20 x) + 12 \ln (7 x^{3} + 10 x^{2}) x$

Step 5.3

Reorder terms.

$(21 x^{2} + 20 x) \frac{6 x^{2}}{7 x^{3} + 10 x^{2}} + 12 x \ln (7 x^{3} + 10 x^{2})$