Calculus Examples

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

Step 1

Rewrite the equation.

$d y = e^{4 - x} d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int d y = \int e^{4 - x} d x$

Step 2.2

Apply the constant rule.

$y + C_{1} = \int e^{4 - x} d x$

Step 2.3

Integrate the right side.

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Step 2.3.1

Let $u = 4 - x$ . Then $d u = - d x$ , so $- d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 2.3.1.1

Let $u = 4 - x$ . Find $\frac{d u}{d x}$ .

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Step 2.3.1.1.1

Differentiate $4 - x$ .

$\frac{d}{d x} [4 - x]$

Step 2.3.1.1.2

Differentiate.

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Step 2.3.1.1.2.1

By the Sum Rule, the derivative of $4 - x$ with respect to $x$ is $\frac{d}{d x} [4] + \frac{d}{d x} [- x]$ .

$\frac{d}{d x} [4] + \frac{d}{d x} [- x]$

Step 2.3.1.1.2.2

Since $4$ is constant with respect to $x$ , the derivative of $4$ with respect to $x$ is $0$ .

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

Step 2.3.1.1.3

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

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Step 2.3.1.1.3.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- x$ with respect to $x$ is $- \frac{d}{d x} [x]$ .

$0 - \frac{d}{d x} [x]$

Step 2.3.1.1.3.2

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

$0 - 1 \cdot 1$

Step 2.3.1.1.3.3

Multiply $- 1$ by $1$ .

$0 - 1$

Step 2.3.1.1.4

Subtract $1$ from $0$ .

$- 1$

Step 2.3.1.2

Rewrite the problem using $u$ and $d u$ .

$y + C_{1} = \int - e^{u} d u$

Step 2.3.2

Since $- 1$ is constant with respect to $u$ , move $- 1$ out of the integral.

$y + C_{1} = - \int e^{u} d u$

Step 2.3.3

The integral of $e^{u}$ with respect to $u$ is $e^{u}$ .

$y + C_{1} = - (e^{u} + C_{2})$

Step 2.3.4

Simplify.

$y + C_{1} = - e^{u} + C_{2}$

Step 2.3.5

Replace all occurrences of $u$ with $4 - x$ .

$y + C_{1} = - e^{4 - x} + C_{2}$

Step 2.4

Group the constant of integration on the right side as $K$ .

$y = - e^{4 - x} + K$