Calculus Examples

$1 - x e^{y}$

Step 1

Differentiate.

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Step 1.1

By the Sum Rule, the derivative of $1 - x e^{y}$ with respect to $x$ is $\frac{d}{d x} [1] + \frac{d}{d x} [- x e^{y}]$ .

$\frac{d}{d x} [1] + \frac{d}{d x} [- x e^{y}]$

Step 1.2

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$0 + \frac{d}{d x} [- x e^{y}]$

Step 2

Evaluate $\frac{d}{d x} [- x e^{y}]$ .

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Step 2.1

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

$0 - e^{y} \frac{d}{d x} [x]$

Step 2.2

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

$0 - e^{y} \cdot 1$

Step 2.3

Multiply $- 1$ by $1$ .

$0 - e^{y}$

Step 3

Subtract $e^{y}$ from $0$ .

$- e^{y}$