Calculus Examples

$z = x^{3} e^{x y}$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d y} (z) = \frac{d}{d y} (x^{3} e^{x y})$

Step 2

The derivative of $z$ with respect to $y$ is $z'$ .

$z'$

Step 3

Differentiate the right side of the equation.

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Step 3.1

Since $x^{3}$ is constant with respect to $y$ , the derivative of $x^{3} e^{x y}$ with respect to $y$ is $x^{3} \frac{d}{d y} [e^{x y}]$ .

$x^{3} \frac{d}{d y} [e^{x y}]$

Step 3.2

Differentiate using the chain rule, which states that $\frac{d}{d y} [f (g (y))]$ is $f' (g (y)) g' (y)$ where $f (y) = e^{y}$ and $g (y) = x y$ .

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Step 3.2.1

To apply the Chain Rule, set $u$ as $x y$ .

$x^{3} (\frac{d}{d u} [e^{u}] \frac{d}{d y} [x y])$

Step 3.2.2

Differentiate using the Exponential Rule which states that $\frac{d}{d u} [a^{u}]$ is $a^{u} \ln (a)$ where $a$ = $e$ .

$x^{3} (e^{u} \frac{d}{d y} [x y])$

Step 3.2.3

Replace all occurrences of $u$ with $x y$ .

$x^{3} (e^{x y} \frac{d}{d y} [x y])$

Step 3.3

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

$x^{3} (e^{x y} (x \frac{d}{d y} [y]))$

Step 3.4

Multiply $x^{3}$ by $x$ by adding the exponents.

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Step 3.4.1

Move $x$ .

$x \cdot x^{3} (e^{x y} (\frac{d}{d y} [y]))$

Step 3.4.2

Multiply $x$ by $x^{3}$ .

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Step 3.4.2.1

Raise $x$ to the power of $1$ .

$x^{1} x^{3} (e^{x y} (\frac{d}{d y} [y]))$

Step 3.4.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$x^{1 + 3} (e^{x y} (\frac{d}{d y} [y]))$

Step 3.4.3

Add $1$ and $3$ .

$x^{4} (e^{x y} (\frac{d}{d y} [y]))$

Step 3.5

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

$x^{4} (e^{x y} \cdot 1)$

Step 3.6

Multiply $e^{x y}$ by $1$ .

$x^{4} e^{x y}$

Step 4

Reform the equation by setting the left side equal to the right side.

$z' = x^{4} e^{x y}$

Step 5

Replace $z'$ with $\frac{d z}{d y}$ .

$\frac{d z}{d y} = x^{4} e^{x y}$