Calculus Examples

$\frac{d y}{d x} = \frac{1}{\sqrt{x}}$ , $f (4) = 1$

Step 1

Rewrite the equation.

$d y = \frac{1}{\sqrt{x}} d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int d y = \int \frac{1}{\sqrt{x}} d x$

Step 2.2

Apply the constant rule.

$y + C_{1} = \int \frac{1}{\sqrt{x}} d x$

Step 2.3

Integrate the right side.

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Step 2.3.1

Apply basic rules of exponents.

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Step 2.3.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x}$ as $x^{\frac{1}{2}}$ .

$y + C_{1} = \int \frac{1}{x^{\frac{1}{2}}} d x$

Step 2.3.1.2

Move $x^{\frac{1}{2}}$ out of the denominator by raising it to the $- 1$ power.

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

Step 2.3.1.3

Multiply the exponents in ${(x^{\frac{1}{2}})}^{- 1}$ .

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Step 2.3.1.3.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$y + C_{1} = \int x^{\frac{1}{2} \cdot - 1} d x$

Step 2.3.1.3.2

Combine $\frac{1}{2}$ and $- 1$ .

$y + C_{1} = \int x^{\frac{- 1}{2}} d x$

Step 2.3.1.3.3

Move the negative in front of the fraction.

$y + C_{1} = \int x^{- \frac{1}{2}} d x$

Step 2.3.2

By the Power Rule, the integral of $x^{- \frac{1}{2}}$ with respect to $x$ is $2 x^{\frac{1}{2}}$ .

$y + C_{1} = 2 x^{\frac{1}{2}} + C_{2}$

Step 2.4

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

$y = 2 x^{\frac{1}{2}} + K$

Step 3

Use the initial condition to find the value of $K$ by substituting $4$ for $x$ and $1$ for $y$ in $y = 2 x^{\frac{1}{2}} + K$ .

$1 = 2 \cdot 4^{\frac{1}{2}} + K$

Step 4

Solve for $K$ .

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Step 4.1

Rewrite the equation as $2 \cdot 4^{\frac{1}{2}} + K = 1$ .

$2 \cdot 4^{\frac{1}{2}} + K = 1$

Step 4.2

Simplify each term.

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Step 4.2.1

Rewrite $4$ as $2^{2}$ .

$2 \cdot {(2^{2})}^{\frac{1}{2}} + K = 1$

Step 4.2.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$2 \cdot 2^{2 (\frac{1}{2})} + K = 1$

Step 4.2.3

Cancel the common factor of $2$ .

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Step 4.2.3.1

Cancel the common factor.

$2 \cdot 2^{2 (\frac{1}{2})} + K = 1$

Step 4.2.3.2

Rewrite the expression.

$2 \cdot 2^{1} + K = 1$

Step 4.2.4

Evaluate the exponent.

$2 \cdot 2 + K = 1$

Step 4.2.5

Multiply $2$ by $2$ .

$4 + K = 1$

Step 4.3

Move all terms not containing $K$ to the right side of the equation.

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Step 4.3.1

Subtract $4$ from both sides of the equation.

$K = 1 - 4$

Step 4.3.2

Subtract $4$ from $1$ .

$K = - 3$

Step 5

Substitute $- 3$ for $K$ in $y = 2 x^{\frac{1}{2}} + K$ and simplify.

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Step 5.1

Substitute $- 3$ for $K$ .

$y = 2 x^{\frac{1}{2}} - 3$