Calculus Examples

$x^{2 y}$

Step 1

Differentiate using the Generalized Power Rule which states that $\frac{d}{d y} [f^{g}]$ is $f^{g} (f' \frac{g}{f} + g' \ln (f))$ where $f = x$ and $g = 2 y$ .

$\frac{d}{d y} [x] (2 y x^{2 y - 1}) + \frac{d}{d y} [2 y] (x^{2 y} \ln (x))$

Step 2

Differentiate.

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Step 2.1

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

$0 (2 y x^{2 y - 1}) + \frac{d}{d y} [2 y] (x^{2 y} \ln (x))$

Step 2.2

Simplify the expression.

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Step 2.2.1

Multiply $2$ by $0$ .

$0 (y x^{2 y - 1}) + \frac{d}{d y} [2 y] (x^{2 y} \ln (x))$

Step 2.2.2

Multiply $y$ by $0$ .

$0 x^{2 y - 1} + \frac{d}{d y} [2 y] (x^{2 y} \ln (x))$

Step 2.2.3

Multiply $0$ by $x^{2 y - 1}$ .

$0 + \frac{d}{d y} [2 y] (x^{2 y} \ln (x))$

Step 2.2.4

Add $0$ and $\frac{d}{d y} [2 y] (x^{2 y} \ln (x))$ .

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

Step 2.3

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

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

Step 2.4

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

$2 \cdot 1 (x^{2 y} \ln (x))$

Step 2.5

Multiply $2$ by $1$ .

$2 x^{2 y} \ln (x)$