Calculus Examples

$2 (6) (0.5) + 2 (8) \frac{d y}{d t} = 0$

Step 1

Separate the variables.

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Step 1.1

Solve for $\frac{d y}{d t}$ .

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Step 1.1.1

Simplify each term.

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Step 1.1.1.1

Multiply $2 (6) \cdot 0.5$ .

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Step 1.1.1.1.1

Multiply $2$ by $6$ .

$12 \cdot 0.5 + 2 (8) \frac{d y}{d t} = 0$

Step 1.1.1.1.2

Multiply $12$ by $0.5$ .

$6 + 2 (8) \frac{d y}{d t} = 0$

$6 + 2 (8) \frac{d y}{d t} = 0$

Step 1.1.1.2

Multiply $2$ by $8$ .

$6 + 16 \frac{d y}{d t} = 0$

$6 + 16 \frac{d y}{d t} = 0$

Step 1.1.2

Subtract $6$ from both sides of the equation.

$16 \frac{d y}{d t} = - 6$

Step 1.1.3

Divide each term in $16 \frac{d y}{d t} = - 6$ by $16$ and simplify.

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Step 1.1.3.1

Divide each term in $16 \frac{d y}{d t} = - 6$ by $16$ .

$\frac{16 \frac{d y}{d t}}{16} = \frac{- 6}{16}$

Step 1.1.3.2

Simplify the left side.

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Step 1.1.3.2.1

Cancel the common factor of $16$ .

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Step 1.1.3.2.1.1

Cancel the common factor.

$\frac{16 \frac{d y}{d t}}{16} = \frac{- 6}{16}$

Step 1.1.3.2.1.2

Divide $\frac{d y}{d t}$ by $1$ .

$\frac{d y}{d t} = \frac{- 6}{16}$

$\frac{d y}{d t} = \frac{- 6}{16}$

$\frac{d y}{d t} = \frac{- 6}{16}$

Step 1.1.3.3

Simplify the right side.

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Step 1.1.3.3.1

Cancel the common factor of $- 6$ and $16$ .

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Step 1.1.3.3.1.1

Factor $2$ out of $- 6$ .

$\frac{d y}{d t} = \frac{2 (- 3)}{16}$

Step 1.1.3.3.1.2

Cancel the common factors.

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Step 1.1.3.3.1.2.1

Factor $2$ out of $16$ .

$\frac{d y}{d t} = \frac{2 \cdot - 3}{2 \cdot 8}$

Step 1.1.3.3.1.2.2

Cancel the common factor.

$\frac{d y}{d t} = \frac{2 \cdot - 3}{2 \cdot 8}$

Step 1.1.3.3.1.2.3

Rewrite the expression.

$\frac{d y}{d t} = \frac{- 3}{8}$

$\frac{d y}{d t} = \frac{- 3}{8}$

$\frac{d y}{d t} = \frac{- 3}{8}$

Step 1.1.3.3.2

Move the negative in front of the fraction.

$\frac{d y}{d t} = - \frac{3}{8}$

$\frac{d y}{d t} = - \frac{3}{8}$

$\frac{d y}{d t} = - \frac{3}{8}$

$\frac{d y}{d t} = - \frac{3}{8}$

Step 1.2

Rewrite the equation.

$d y = - \frac{3}{8} d t$

$d y = - \frac{3}{8} d t$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int d y = \int - \frac{3}{8} d t$

Step 2.2

Apply the constant rule.

$y + C_{1} = \int - \frac{3}{8} d t$

Step 2.3

Apply the constant rule.

$y + C_{1} = - \frac{3}{8} t + C_{2}$

Step 2.4

Group the constant of integration on the right side as $K$ .

$y = - \frac{3}{8} t + K$

$y = - \frac{3}{8} t + K$

$[\begin{array}{c} x^{2} & \frac{1}{2} \\ \sqrt{π} & \int x d x \end{array}]$