Calculus Examples

$\frac{d y}{d x} = - 3 \sec^{2} (x - 7)$ , $y (7) = 7$

Step 1

Rewrite the equation.

$d y = - 3 \sec^{2} (x - 7) d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int d y = \int - 3 \sec^{2} (x - 7) d x$

Step 2.2

Apply the constant rule.

$y + C_{1} = \int - 3 \sec^{2} (x - 7) d x$

Step 2.3

Integrate the right side.

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Step 2.3.1

Since $- 3$ is constant with respect to $x$ , move $- 3$ out of the integral.

$y + C_{1} = - 3 \int \sec^{2} (x - 7) d x$

Step 2.3.2

Let $u = x - 7$ . Then $d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 2.3.2.1

Let $u = x - 7$ . Find $\frac{d u}{d x}$ .

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Step 2.3.2.1.1

Differentiate $x - 7$ .

$\frac{d}{d x} [x - 7]$

Step 2.3.2.1.2

By the Sum Rule, the derivative of $x - 7$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [- 7]$ .

$\frac{d}{d x} [x] + \frac{d}{d x} [- 7]$

Step 2.3.2.1.3

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

$1 + \frac{d}{d x} [- 7]$

Step 2.3.2.1.4

Since $- 7$ is constant with respect to $x$ , the derivative of $- 7$ with respect to $x$ is $0$ .

$1 + 0$

Step 2.3.2.1.5

Add $1$ and $0$ .

$1$

Step 2.3.2.2

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

$y + C_{1} = - 3 \int \sec^{2} (u) d u$

Step 2.3.3

Since the derivative of $\tan (u)$ is $\sec^{2} (u)$ , the integral of $\sec^{2} (u)$ is $\tan (u)$ .

$y + C_{1} = - 3 (\tan (u) + C_{2})$

Step 2.3.4

Simplify.

$y + C_{1} = - 3 \tan (u) + C_{2}$

Step 2.3.5

Replace all occurrences of $u$ with $x - 7$ .

$y + C_{1} = - 3 \tan (x - 7) + C_{2}$

Step 2.4

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

$y = - 3 \tan (x - 7) + K$

Step 3

Use the initial condition to find the value of $K$ by substituting $7$ for $x$ and $7$ for $y$ in $y = - 3 \tan (x - 7) + K$ .

$7 = - 3 \tan (7 - 7) + K$

Step 4

Solve for $K$ .

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Step 4.1

Rewrite the equation as $- 3 \tan (7 - 7) + K = 7$ .

$- 3 \tan (7 - 7) + K = 7$

Step 4.2

Simplify the left side.

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Step 4.2.1

Simplify $- 3 \tan (7 - 7) + K$ .

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Step 4.2.1.1

Subtract $7$ from $7$ .

$- 3 \tan (0) + K = 7$

Step 4.2.1.2

Simplify each term.

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Step 4.2.1.2.1

The exact value of $\tan (0)$ is $0$ .

$- 3 \cdot 0 + K = 7$

Step 4.2.1.2.2

Multiply $- 3$ by $0$ .

$0 + K = 7$

Step 4.2.1.3

Add $0$ and $K$ .

$K = 7$

Step 5

Substitute $7$ for $K$ in $y = - 3 \tan (x - 7) + K$ and simplify.

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Step 5.1

Substitute $7$ for $K$ .

$y = - 3 \tan (x - 7) + 7$