Calculus Examples

$y \cos (x) = x^{2} + y^{2}$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (y \cos (x)) = \frac{d}{d x} (x^{2} + y^{2})$

Step 2

Differentiate the left side of the equation.

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Step 2.1

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

$y \frac{d}{d x} [\cos (x)] + \cos (x) \frac{d}{d x} [y]$

Step 2.2

The derivative of $\cos (x)$ with respect to $x$ is $- \sin (x)$ .

$y (- \sin (x)) + \cos (x) \frac{d}{d x} [y]$

Step 2.3

Rewrite $\frac{d}{d x} [y]$ as $y'$ .

$y (- \sin (x)) + \cos (x) y'$

Step 2.4

Reorder terms.

$- y \sin (x) + \cos (x) y'$

Step 3

Differentiate the right side of the equation.

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Step 3.1

Differentiate.

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Step 3.1.1

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

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

Step 3.1.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

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

Step 3.2

Evaluate $\frac{d}{d x} [y^{2}]$ .

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Step 3.2.1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{2}$ and $g (x) = y$ .

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Step 3.2.1.1

To apply the Chain Rule, set $u$ as $y$ .

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

Step 3.2.1.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = 2$ .

$2 x + 2 u \frac{d}{d x} [y]$

Step 3.2.1.3

Replace all occurrences of $u$ with $y$ .

$2 x + 2 y \frac{d}{d x} [y]$

Step 3.2.2

Rewrite $\frac{d}{d x} [y]$ as $y'$ .

$2 x + 2 y y'$

Step 3.3

Reorder terms.

$2 y y' + 2 x$

Step 4

Reform the equation by setting the left side equal to the right side.

$- y \sin (x) + \cos (x) y' = 2 y y' + 2 x$

Step 5

Solve for $y'$ .

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Step 5.1

Simplify the left side.

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Step 5.1.1

Reorder factors in $- y \sin (x) + \cos (x) y'$ .

$- y \sin (x) + y' \cos (x) = 2 y y' + 2 x$

Step 5.2

Subtract $2 y y'$ from both sides of the equation.

$- y \sin (x) + y' \cos (x) - 2 y y' = 2 x$

Step 5.3

Add $y \sin (x)$ to both sides of the equation.

$y' \cos (x) - 2 y y' = 2 x + y \sin (x)$

Step 5.4

Factor $y'$ out of $y' \cos (x) - 2 y y'$ .

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Step 5.4.1

Factor $y'$ out of $y' \cos (x)$ .

$y' (\cos (x)) - 2 y y' = 2 x + y \sin (x)$

Step 5.4.2

Factor $y'$ out of $- 2 y y'$ .

$y' (\cos (x)) + y' (- 2 y) = 2 x + y \sin (x)$

Step 5.4.3

Factor $y'$ out of $y' (\cos (x)) + y' (- 2 y)$ .

$y' (\cos (x) - 2 y) = 2 x + y \sin (x)$

Step 5.5

Divide each term in $y' (\cos (x) - 2 y) = 2 x + y \sin (x)$ by $\cos (x) - 2 y$ and simplify.

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Step 5.5.1

Divide each term in $y' (\cos (x) - 2 y) = 2 x + y \sin (x)$ by $\cos (x) - 2 y$ .

$\frac{y' (\cos (x) - 2 y)}{\cos (x) - 2 y} = \frac{2 x}{\cos (x) - 2 y} + \frac{y \sin (x)}{\cos (x) - 2 y}$

Step 5.5.2

Simplify the left side.

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Step 5.5.2.1

Cancel the common factor of $\cos (x) - 2 y$ .

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Step 5.5.2.1.1

Cancel the common factor.

$\frac{y' (\cos (x) - 2 y)}{\cos (x) - 2 y} = \frac{2 x}{\cos (x) - 2 y} + \frac{y \sin (x)}{\cos (x) - 2 y}$

Step 5.5.2.1.2

Divide $y'$ by $1$ .

$y' = \frac{2 x}{\cos (x) - 2 y} + \frac{y \sin (x)}{\cos (x) - 2 y}$

Step 5.5.3

Simplify the right side.

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Step 5.5.3.1

Combine the numerators over the common denominator.

$y' = \frac{2 x + y \sin (x)}{\cos (x) - 2 y}$

Step 6

Replace $y'$ with $\frac{d y}{d x}$ .

$\frac{d y}{d x} = \frac{2 x + y \sin (x)}{\cos (x) - 2 y}$