Calculus Examples

$\frac{d^{4} y}{d x^{4}}$ , $y = 6 x^{6} - 6 x^{4} + 3 x^{2}$

Step 1

Replace the variable $y$ with $6 x^{6} - 6 x^{4} + 3 x^{2}$ in the expression.

$\frac{d^{4} (6 x^{6} - 6 x^{4} + 3 x^{2})}{d x^{4}}$

Step 2

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

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Step 2.1

Factor $d$ out of $d^{4} (6 x^{6} - 6 x^{4} + 3 x^{2})$ .

$\frac{d (d^{3} (6 x^{6} - 6 x^{4} + 3 x^{2}))}{d x^{4}}$

Step 2.2

Cancel the common factors.

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Step 2.2.1

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

$\frac{d (d^{3} (6 x^{6} - 6 x^{4} + 3 x^{2}))}{d (x^{4})}$

Step 2.2.2

Cancel the common factor.

$\frac{d (d^{3} (6 x^{6} - 6 x^{4} + 3 x^{2}))}{d x^{4}}$

Step 2.2.3

Rewrite the expression.

$\frac{d^{3} (6 x^{6} - 6 x^{4} + 3 x^{2})}{x^{4}}$

Step 3

Factor $3 x^{2}$ out of $6 x^{6} - 6 x^{4} + 3 x^{2}$ .

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Step 3.1

Factor $3 x^{2}$ out of $6 x^{6}$ .

$\frac{d^{3} (3 x^{2} (2 x^{4}) - 6 x^{4} + 3 x^{2})}{x^{4}}$

Step 3.2

Factor $3 x^{2}$ out of $- 6 x^{4}$ .

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

Step 3.3

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

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

Step 3.4

Factor $3 x^{2}$ out of $3 x^{2} (2 x^{4}) + 3 x^{2} (- 2 x^{2})$ .

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

Step 3.5

Factor $3 x^{2}$ out of $3 x^{2} (2 x^{4} - 2 x^{2}) + 3 x^{2} (1)$ .

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

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

Step 4

Cancel the common factor of $x^{2}$ and $x^{4}$ .

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Step 4.1

Factor $x^{2}$ out of $d^{3} \cdot 3 x^{2} (2 x^{4} - 2 x^{2} + 1)$ .

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

Step 4.2

Cancel the common factors.

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Step 4.2.1

Factor $x^{2}$ out of $x^{4}$ .

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

Step 4.2.2

Cancel the common factor.

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

Step 4.2.3

Rewrite the expression.

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

Step 5

Move $3$ to the left of $d^{3}$ .

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