Calculus Examples

$\frac{d^{2} y}{d x^{2}} = 0$

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 $x$ .

$\frac{d y}{d x} = \int 0 d x$

Step 1.2

The integral of $0$ with respect to $x$ is $0$ .

$\frac{d y}{d x} = 0 + C_{1}$

Step 1.3

Add $0$ and $C_{1}$ .

$\frac{d y}{d x} = C_{1}$

Step 2

Rewrite the equation.

$d y = C_{1} d x$

Step 3

Integrate both sides.

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Step 3.1

Set up an integral on each side.

$\int d y = \int C_{1} d x$

Step 3.2

Apply the constant rule.

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

Step 3.3

Apply the constant rule.

$y + C_{2} = C_{1} x + C_{3}$

Step 3.4

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

$y = C x + D$