Calculus Examples

$f (y) = (y + 2) e^{x (y + 2)}$

Step 1

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

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

Step 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 $g (y) = x (y + 2)$ .

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Step 2.1

To apply the Chain Rule, set $u$ as $x (y + 2)$ .

$(y + 2) (\frac{d}{d u} [e^{u}] \frac{d}{d y} [x (y + 2)]) + e^{x (y + 2)} \frac{d}{d y} [y + 2]$

Step 2.2

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

$(y + 2) (e^{u} \frac{d}{d y} [x (y + 2)]) + e^{x (y + 2)} \frac{d}{d y} [y + 2]$

Step 2.3

Replace all occurrences of $u$ with $x (y + 2)$ .

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

Step 3

Differentiate.

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Step 3.1

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

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

Step 3.2

By the Sum Rule, the derivative of $y + 2$ with respect to $y$ is $\frac{d}{d y} [y] + \frac{d}{d y} [2]$ .

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

Step 3.3

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

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

Step 3.4

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

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

Step 3.5

Simplify the expression.

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Step 3.5.1

Add $1$ and $0$ .

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

Step 3.5.2

Multiply $x$ by $1$ .

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

Step 3.6

By the Sum Rule, the derivative of $y + 2$ with respect to $y$ is $\frac{d}{d y} [y] + \frac{d}{d y} [2]$ .

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

Step 3.7

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

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

Step 3.8

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

$(y + 2) e^{x (y + 2)} x + e^{x (y + 2)} (1 + 0)$

Step 3.9

Simplify the expression.

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Step 3.9.1

Add $1$ and $0$ .

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

Step 3.9.2

Multiply $e^{x (y + 2)}$ by $1$ .

$(y + 2) e^{x (y + 2)} x + e^{x (y + 2)}$

Step 4

Simplify.

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Step 4.1

Apply the distributive property.

$(y + 2) e^{x y + x \cdot 2} x + e^{x (y + 2)}$

Step 4.2

Apply the distributive property.

$(y + 2) e^{x y + x \cdot 2} x + e^{x y + x \cdot 2}$

Step 4.3

Combine terms.

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Step 4.3.1

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

$(y + 2) e^{x y + 2 \cdot x} x + e^{x y + x \cdot 2}$

Step 4.3.2

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

$(y + 2) e^{x y + 2 x} x + e^{x y + 2 x}$

Step 4.4

Reorder terms.

$e^{x y + 2 x} x (y + 2) + e^{x y + 2 x}$