Calculus Examples

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

Step 1

Find the first derivative.

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Step 1.1

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

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Step 1.1.1

Factor $d$ out of $d^{2} y$ .

$\frac{d}{d \cdot d} [\frac{d (d y)}{d x^{2}}]$

Step 1.1.2

Cancel the common factors.

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Step 1.1.2.1

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

$\frac{d}{d \cdot d} [\frac{d (d y)}{d (x^{2})}]$

Step 1.1.2.2

Cancel the common factor.

$\frac{d}{d \cdot d} [\frac{d (d y)}{d x^{2}}]$

Step 1.1.2.3

Rewrite the expression.

$\frac{d}{d \cdot d} [\frac{d y}{x^{2}}]$

Step 1.2

Since $\frac{y}{x^{2}}$ is constant with respect to $d$ , the derivative of $\frac{d y}{x^{2}}$ with respect to $d$ is $\frac{y}{x^{2}} \frac{d}{d \cdot d} [d]$ .

$\frac{y}{x^{2}} \frac{d}{d \cdot d} [d]$

Step 1.3

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

$\frac{y}{x^{2}} \cdot 1$

Step 1.4

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

$f' (d) = \frac{y}{x^{2}}$

Step 2

Since $\frac{y}{x^{2}}$ is constant with respect to $d$ , the derivative of $\frac{y}{x^{2}}$ with respect to $d$ is $0$ .

$f'' (d) = 0$