Calculus Examples

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

Step 1

Integrate both sides with respect to $x$ .

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Step 1.1

The first derivative is equal to the integral of the second derivative with respect to $x$ .

$\frac{d y}{d x} = \int - \frac{3}{x^{4}} d x$

Step 1.2

Since $- 1$ is constant with respect to $x$ , move $- 1$ out of the integral.

$\frac{d y}{d x} = - \int \frac{3}{x^{4}} d x$

Step 1.3

Since $3$ is constant with respect to $x$ , move $3$ out of the integral.

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

Step 1.4

Simplify the expression.

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Step 1.4.1

Multiply $3$ by $- 1$ .

$\frac{d y}{d x} = - 3 \int \frac{1}{x^{4}} d x$

Step 1.4.2

Move $x^{4}$ out of the denominator by raising it to the $- 1$ power.

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

Step 1.4.3

Multiply the exponents in ${(x^{4})}^{- 1}$ .

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Step 1.4.3.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{d y}{d x} = - 3 \int x^{4 \cdot - 1} d x$

Step 1.4.3.2

Multiply $4$ by $- 1$ .

$\frac{d y}{d x} = - 3 \int x^{- 4} d x$

Step 1.5

By the Power Rule, the integral of $x^{- 4}$ with respect to $x$ is $- \frac{1}{3} x^{- 3}$ .

$\frac{d y}{d x} = - 3 (- \frac{1}{3} x^{- 3} + C_{1})$

Step 1.6

Simplify the answer.

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Step 1.6.1

Simplify.

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Step 1.6.1.1

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

$\frac{d y}{d x} = - 3 (- \frac{x^{- 3}}{3} + C_{1})$

Step 1.6.1.2

Move $x^{- 3}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{d y}{d x} = - 3 (- \frac{1}{3 x^{3}} + C_{1})$

Step 1.6.2

Simplify.

$\frac{d y}{d x} = - 3 (- \frac{1}{3 x^{3}}) + C_{1}$

Step 1.6.3

Simplify.

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Step 1.6.3.1

Multiply $- 1$ by $- 3$ .

$\frac{d y}{d x} = 3 \frac{1}{3 x^{3}} + C_{1}$

Step 1.6.3.2

Combine $3$ and $\frac{1}{3 x^{3}}$ .

$\frac{d y}{d x} = \frac{3}{3 x^{3}} + C_{1}$

Step 1.6.3.3

Cancel the common factor of $3$ .

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Step 1.6.3.3.1

Cancel the common factor.

$\frac{d y}{d x} = \frac{3}{3 x^{3}} + C_{1}$

Step 1.6.3.3.2

Rewrite the expression.

$\frac{d y}{d x} = \frac{1}{x^{3}} + C_{1}$

Step 2

Rewrite the equation.

$d y = (\frac{1}{x^{3}} + C_{1}) d x$

Step 3

Integrate both sides.

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Step 3.1

Set up an integral on each side.

$\int d y = \int \frac{1}{x^{3}} + C_{1} d x$

Step 3.2

Apply the constant rule.

$y + C_{2} = \int \frac{1}{x^{3}} + C_{1} d x$

Step 3.3

Integrate the right side.

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Step 3.3.1

Split the single integral into multiple integrals.

$y + C_{2} = \int \frac{1}{x^{3}} d x + \int C_{1} d x$

Step 3.3.2

Apply basic rules of exponents.

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Step 3.3.2.1

Move $x^{3}$ out of the denominator by raising it to the $- 1$ power.

$y + C_{2} = \int {(x^{3})}^{- 1} d x + \int C_{1} d x$

Step 3.3.2.2

Multiply the exponents in ${(x^{3})}^{- 1}$ .

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Step 3.3.2.2.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$y + C_{2} = \int x^{3 \cdot - 1} d x + \int C_{1} d x$

Step 3.3.2.2.2

Multiply $3$ by $- 1$ .

$y + C_{2} = \int x^{- 3} d x + \int C_{1} d x$

Step 3.3.3

By the Power Rule, the integral of $x^{- 3}$ with respect to $x$ is $- \frac{1}{2} x^{- 2}$ .

$y + C_{2} = - \frac{1}{2} x^{- 2} + C_{3} + \int C_{1} d x$

Step 3.3.4

Apply the constant rule.

$y + C_{2} = - \frac{1}{2} x^{- 2} + C_{3} + C_{1} x + C_{4}$

Step 3.3.5

Simplify.

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Step 3.3.5.1

Simplify.

$y + C_{2} = - \frac{1}{2} \cdot \frac{1}{x^{2}} + C_{1} x + C_{5}$

Step 3.3.5.2

Simplify.

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Step 3.3.5.2.1

Multiply $\frac{1}{x^{2}}$ by $\frac{1}{2}$ .

$y + C_{2} = - \frac{1}{x^{2} \cdot 2} + C_{1} x + C_{5}$

Step 3.3.5.2.2

Move $2$ to the left of $x^{2}$ .

$y + C_{2} = - \frac{1}{2 x^{2}} + C_{1} x + C_{5}$

Step 3.3.6

Reorder terms.

$y + C_{2} = C_{1} x - \frac{1}{2 x^{2}} + C_{5}$

Step 3.4

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

$y = C x - \frac{1}{2 x^{2}} + D$