Calculus Examples

$x^{7} + 8 x^{7} \ln (x)$

Step 1

Differentiate.

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Step 1.1

By the Sum Rule, the derivative of $x^{7} + 8 x^{7} \ln (x)$ with respect to $x$ is $\frac{d}{d x} [x^{7}] + \frac{d}{d x} [8 x^{7} \ln (x)]$ .

$\frac{d}{d x} [x^{7}] + \frac{d}{d x} [8 x^{7} \ln (x)]$

Step 1.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 7$ .

$7 x^{6} + \frac{d}{d x} [8 x^{7} \ln (x)]$

Step 2

Evaluate $\frac{d}{d x} [8 x^{7} \ln (x)]$ .

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Step 2.1

Since $8$ is constant with respect to $x$ , the derivative of $8 x^{7} \ln (x)$ with respect to $x$ is $8 \frac{d}{d x} [x^{7} \ln (x)]$ .

$7 x^{6} + 8 \frac{d}{d x} [x^{7} \ln (x)]$

Step 2.2

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 $f (x) = x^{7}$ and $g (x) = \ln (x)$ .

$7 x^{6} + 8 (x^{7} \frac{d}{d x} [\ln (x)] + \ln (x) \frac{d}{d x} [x^{7}])$

Step 2.3

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

$7 x^{6} + 8 (x^{7} \frac{1}{x} + \ln (x) \frac{d}{d x} [x^{7}])$

Step 2.4

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 7$ .

$7 x^{6} + 8 (x^{7} \frac{1}{x} + \ln (x) (7 x^{6}))$

Step 2.5

Combine $x^{7}$ and $\frac{1}{x}$ .

$7 x^{6} + 8 (\frac{x^{7}}{x} + \ln (x) (7 x^{6}))$

Step 2.6

Cancel the common factor of $x^{7}$ and $x$ .

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Step 2.6.1

Factor $x$ out of $x^{7}$ .

$7 x^{6} + 8 (\frac{x \cdot x^{6}}{x} + \ln (x) (7 x^{6}))$

Step 2.6.2

Cancel the common factors.

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Step 2.6.2.1

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

$7 x^{6} + 8 (\frac{x \cdot x^{6}}{x^{1}} + \ln (x) (7 x^{6}))$

Step 2.6.2.2

Factor $x$ out of $x^{1}$ .

$7 x^{6} + 8 (\frac{x \cdot x^{6}}{x \cdot 1} + \ln (x) (7 x^{6}))$

Step 2.6.2.3

Cancel the common factor.

$7 x^{6} + 8 (\frac{x \cdot x^{6}}{x \cdot 1} + \ln (x) (7 x^{6}))$

Step 2.6.2.4

Rewrite the expression.

$7 x^{6} + 8 (\frac{x^{6}}{1} + \ln (x) (7 x^{6}))$

Step 2.6.2.5

Divide $x^{6}$ by $1$ .

$7 x^{6} + 8 (x^{6} + \ln (x) (7 x^{6}))$

Step 3

Simplify.

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Step 3.1

Apply the distributive property.

$7 x^{6} + 8 x^{6} + 8 (\ln (x) (7 x^{6}))$

Step 3.2

Combine terms.

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Step 3.2.1

Multiply $7$ by $8$ .

$7 x^{6} + 8 x^{6} + 56 (\ln (x) (x^{6}))$

Step 3.2.2

Add $7 x^{6}$ and $8 x^{6}$ .

$15 x^{6} + 56 \ln (x) x^{6}$

Step 3.3

Reorder terms.

$15 x^{6} + 56 x^{6} \ln (x)$