Find the Derivative - d/dx x^7 natural log of x-1/5x^5

$x^{7} \ln (x) - \frac{1}{5} x^{5}$

By the Sum Rule, the derivative of $x^{7} \ln (x) - \frac{1}{5} x^{5}$ with respect to $x$ is $\frac{d}{d x} [x^{7} \ln (x)] + \frac{d}{d x} [- \frac{1}{5} x^{5}]$ .

$\frac{d}{d x} [x^{7} \ln (x)] + \frac{d}{d x} [- \frac{1}{5} x^{5}]$

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

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 $g (x) = \ln (x)$ .

$x^{7} \frac{d}{d x} [\ln (x)] + \ln (x) \frac{d}{d x} [x^{7}] + \frac{d}{d x} [- \frac{1}{5} x^{5}]$

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

$x^{7} \frac{1}{x} + \ln (x) \frac{d}{d x} [x^{7}] + \frac{d}{d x} [- \frac{1}{5} x^{5}]$

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

$x^{7} \frac{1}{x} + \ln (x) (7 x^{6}) + \frac{d}{d x} [- \frac{1}{5} x^{5}]$

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

$\frac{x^{7}}{x} + \ln (x) (7 x^{6}) + \frac{d}{d x} [- \frac{1}{5} x^{5}]$

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

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

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

Cancel the common factors.

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

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

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

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

Cancel the common factor.

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

Rewrite the expression.

$\frac{x^{6}}{1} + \ln (x) (7 x^{6}) + \frac{d}{d x} [- \frac{1}{5} x^{5}]$

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

$x^{6} + \ln (x) (7 x^{6}) + \frac{d}{d x} [- \frac{1}{5} x^{5}]$

Evaluate $\frac{d}{d x} [- \frac{1}{5} x^{5}]$ .

Since $- \frac{1}{5}$ is constant with respect to $x$ , the derivative of $- \frac{1}{5} x^{5}$ with respect to $x$ is $- \frac{1}{5} \frac{d}{d x} [x^{5}]$ .

$x^{6} + \ln (x) (7 x^{6}) - \frac{1}{5} \frac{d}{d x} [x^{5}]$

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 5$ .

$x^{6} + \ln (x) (7 x^{6}) - \frac{1}{5} (5 x^{4})$

Multiply $5$ by $- 1$ .

$x^{6} + \ln (x) (7 x^{6}) - 5 (\frac{1}{5}) x^{4}$

Combine $- 5$ and $\frac{1}{5}$ .

$x^{6} + \ln (x) (7 x^{6}) + \frac{- 5}{5} x^{4}$

Combine $\frac{- 5}{5}$ and $x^{4}$ .

$x^{6} + \ln (x) (7 x^{6}) + \frac{- 5 x^{4}}{5}$

Cancel the common factor of $- 5$ and $5$ .

Factor $5$ out of $- 5 x^{4}$ .

$x^{6} + \ln (x) (7 x^{6}) + \frac{5 (- x^{4})}{5}$

Cancel the common factors.

Factor $5$ out of $5$ .

$x^{6} + \ln (x) (7 x^{6}) + \frac{5 (- x^{4})}{5 (1)}$

Cancel the common factor.

$x^{6} + \ln (x) (7 x^{6}) + \frac{5 (- x^{4})}{5 \cdot 1}$

Rewrite the expression.

$x^{6} + \ln (x) (7 x^{6}) + \frac{- x^{4}}{1}$

Divide $- x^{4}$ by $1$ .

$x^{6} + \ln (x) (7 x^{6}) - x^{4}$

Reorder terms.

$x^{6} - x^{4} + 7 x^{6} \ln (x)$

Find the Derivative - d/dx x^7 natural log of x-1/5x^5

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