Calculus Examples

$x^{2} y$

Step 1

Find the first derivative.

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Step 1.1

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

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

Step 1.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$y (2 x)$

Step 1.3

Reorder.

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Step 1.3.1

Move $2$ to the left of $y$ .

$2 \cdot y x$

Step 1.3.2

Reorder the factors of $2 y x$ .

$f' (x) = 2 x y$

Step 2

Find the second derivative.

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Step 2.1

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

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

Step 2.2

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

$2 y \cdot 1$

Step 2.3

Multiply $2$ by $1$ .

$f'' (x) = 2 y$

Step 3

Since $2 y$ is constant with respect to $x$ , the derivative of $2 y$ with respect to $x$ is $0$ .

$f''' (x) = 0$

Step 4

Since $0$ is constant with respect to $x$ , the derivative of $0$ with respect to $x$ is $0$ .

$f^{4} (x) = 0$