Calculus Examples

$y = x^{4} \ln (x) - \frac{1}{3} x^{3}$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (y) = \frac{d}{d x} (x^{4} \ln (x) - \frac{1}{3} x^{3})$

Step 2

The derivative of $y$ with respect to $x$ is $y'$ .

$y'$

Step 3

Differentiate the right side of the equation.

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Step 3.1

By the Sum Rule, the derivative of $x^{4} \ln (x) - \frac{1}{3} x^{3}$ with respect to $x$ is $\frac{d}{d x} [x^{4} \ln (x)] + \frac{d}{d x} [- \frac{1}{3} x^{3}]$ .

$\frac{d}{d x} [x^{4} \ln (x)] + \frac{d}{d x} [- \frac{1}{3} x^{3}]$

Step 3.2

Evaluate $\frac{d}{d x} [x^{4} \ln (x)]$ .

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Step 3.2.1

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 $f (x) = x^{4}$ and $g (x) = \ln (x)$ .

$x^{4} \frac{d}{d x} [\ln (x)] + \ln (x) \frac{d}{d x} [x^{4}] + \frac{d}{d x} [- \frac{1}{3} x^{3}]$

Step 3.2.2

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

$x^{4} \frac{1}{x} + \ln (x) \frac{d}{d x} [x^{4}] + \frac{d}{d x} [- \frac{1}{3} x^{3}]$

Step 3.2.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 4$ .

$x^{4} \frac{1}{x} + \ln (x) (4 x^{3}) + \frac{d}{d x} [- \frac{1}{3} x^{3}]$

Step 3.2.4

Combine $x^{4}$ and $\frac{1}{x}$ .

$\frac{x^{4}}{x} + \ln (x) (4 x^{3}) + \frac{d}{d x} [- \frac{1}{3} x^{3}]$

Step 3.2.5

Cancel the common factor of $x^{4}$ and $x$ .

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Step 3.2.5.1

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

$\frac{x \cdot x^{3}}{x} + \ln (x) (4 x^{3}) + \frac{d}{d x} [- \frac{1}{3} x^{3}]$

Step 3.2.5.2

Cancel the common factors.

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Step 3.2.5.2.1

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

$\frac{x \cdot x^{3}}{x^{1}} + \ln (x) (4 x^{3}) + \frac{d}{d x} [- \frac{1}{3} x^{3}]$

Step 3.2.5.2.2

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

$\frac{x \cdot x^{3}}{x \cdot 1} + \ln (x) (4 x^{3}) + \frac{d}{d x} [- \frac{1}{3} x^{3}]$

Step 3.2.5.2.3

Cancel the common factor.

$\frac{x \cdot x^{3}}{x \cdot 1} + \ln (x) (4 x^{3}) + \frac{d}{d x} [- \frac{1}{3} x^{3}]$

Step 3.2.5.2.4

Rewrite the expression.

$\frac{x^{3}}{1} + \ln (x) (4 x^{3}) + \frac{d}{d x} [- \frac{1}{3} x^{3}]$

Step 3.2.5.2.5

Divide $x^{3}$ by $1$ .

$x^{3} + \ln (x) (4 x^{3}) + \frac{d}{d x} [- \frac{1}{3} x^{3}]$

Step 3.3

Evaluate $\frac{d}{d x} [- \frac{1}{3} x^{3}]$ .

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Step 3.3.1

Since $- \frac{1}{3}$ is constant with respect to $x$ , the derivative of $- \frac{1}{3} x^{3}$ with respect to $x$ is $- \frac{1}{3} \frac{d}{d x} [x^{3}]$ .

$x^{3} + \ln (x) (4 x^{3}) - \frac{1}{3} \frac{d}{d x} [x^{3}]$

Step 3.3.2

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

$x^{3} + \ln (x) (4 x^{3}) - \frac{1}{3} (3 x^{2})$

Step 3.3.3

Multiply $3$ by $- 1$ .

$x^{3} + \ln (x) (4 x^{3}) - 3 (\frac{1}{3}) x^{2}$

Step 3.3.4

Combine $- 3$ and $\frac{1}{3}$ .

$x^{3} + \ln (x) (4 x^{3}) + \frac{- 3}{3} x^{2}$

Step 3.3.5

Combine $\frac{- 3}{3}$ and $x^{2}$ .

$x^{3} + \ln (x) (4 x^{3}) + \frac{- 3 x^{2}}{3}$

Step 3.3.6

Cancel the common factor of $- 3$ and $3$ .

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Step 3.3.6.1

Factor $3$ out of $- 3 x^{2}$ .

$x^{3} + \ln (x) (4 x^{3}) + \frac{3 (- x^{2})}{3}$

Step 3.3.6.2

Cancel the common factors.

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Step 3.3.6.2.1

Factor $3$ out of $3$ .

$x^{3} + \ln (x) (4 x^{3}) + \frac{3 (- x^{2})}{3 (1)}$

Step 3.3.6.2.2

Cancel the common factor.

$x^{3} + \ln (x) (4 x^{3}) + \frac{3 (- x^{2})}{3 \cdot 1}$

Step 3.3.6.2.3

Rewrite the expression.

$x^{3} + \ln (x) (4 x^{3}) + \frac{- x^{2}}{1}$

Step 3.3.6.2.4

Divide $- x^{2}$ by $1$ .

$x^{3} + \ln (x) (4 x^{3}) - x^{2}$

Step 3.4

Reorder terms.

$x^{3} - x^{2} + 4 x^{3} \ln (x)$

Step 4

Reform the equation by setting the left side equal to the right side.

$y' = x^{3} - x^{2} + 4 x^{3} \ln (x)$

Step 5

Replace $y'$ with $\frac{d y}{d x}$ .

$\frac{d y}{d x} = x^{3} - x^{2} + 4 x^{3} \ln (x)$