Calculus Examples

$y \cos (x) = 3 x^{2} + 5 y^{2}$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (y \cos (x)) = \frac{d}{d x} (3 x^{2} + 5 y^{2})$

Step 2

Differentiate the left side of the equation.

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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 $g (x) = \cos (x)$ .

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

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

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

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

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

Reorder terms.

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

Step 3

Differentiate the right side of the equation.

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By the Sum Rule, the derivative of $3 x^{2} + 5 y^{2}$ with respect to $x$ is $\frac{d}{d x} [3 x^{2}] + \frac{d}{d x} [5 y^{2}]$ .

$\frac{d}{d x} [3 x^{2}] + \frac{d}{d x} [5 y^{2}]$

Evaluate $\frac{d}{d x} [3 x^{2}]$ .

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Since $3$ is constant with respect to $x$ , the derivative of $3 x^{2}$ with respect to $x$ is $3 \frac{d}{d x} [x^{2}]$ .

$3 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [5 y^{2}]$

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

$3 (2 x) + \frac{d}{d x} [5 y^{2}]$

Multiply $2$ by $3$ .

$6 x + \frac{d}{d x} [5 y^{2}]$

Evaluate $\frac{d}{d x} [5 y^{2}]$ .

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Since $5$ is constant with respect to $x$ , the derivative of $5 y^{2}$ with respect to $x$ is $5 \frac{d}{d x} [y^{2}]$ .

$6 x + 5 \frac{d}{d x} [y^{2}]$

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

$6 x + 5 (\frac{d}{d u} [u^{2}] \frac{d}{d x} [y])$

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

$6 x + 5 (2 u \frac{d}{d x} [y])$

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

$6 x + 5 (2 y \frac{d}{d x} [y])$

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

$6 x + 5 (2 y y')$

Multiply $2$ by $5$ .

$6 x + 10 y y'$

Reorder terms.

$10 y y' + 6 x$

Step 4

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

$- y \sin (x) + \cos (x) y' = 10 y y' + 6 x$

Step 5

Solve for $y'$ .

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Simplify the left side.

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

$- y \sin (x) + y' \cos (x) = 10 y y' + 6 x$

Subtract $10 y y'$ from both sides of the equation.

$- y \sin (x) + y' \cos (x) - 10 y y' = 6 x$

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

$y' \cos (x) - 10 y y' = 6 x + y \sin (x)$

Factor $y'$ out of $y' \cos (x) - 10 y y'$ .

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

$y' (\cos (x)) - 10 y y' = 6 x + y \sin (x)$

Factor $y'$ out of $- 10 y y'$ .

$y' (\cos (x)) + y' (- 10 y) = 6 x + y \sin (x)$

Factor $y'$ out of $y' (\cos (x)) + y' (- 10 y)$ .

$y' (\cos (x) - 10 y) = 6 x + y \sin (x)$

Divide each term in $y' (\cos (x) - 10 y) = 6 x + y \sin (x)$ by $\cos (x) - 10 y$ and simplify.

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Divide each term in $y' (\cos (x) - 10 y) = 6 x + y \sin (x)$ by $\cos (x) - 10 y$ .

$\frac{y' (\cos (x) - 10 y)}{\cos (x) - 10 y} = \frac{6 x}{\cos (x) - 10 y} + \frac{y \sin (x)}{\cos (x) - 10 y}$

Simplify the left side.

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Cancel the common factor of $\cos (x) - 10 y$ .

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Cancel the common factor.

$\frac{y' (\cos (x) - 10 y)}{\cos (x) - 10 y} = \frac{6 x}{\cos (x) - 10 y} + \frac{y \sin (x)}{\cos (x) - 10 y}$

Divide $y'$ by $1$ .

$y' = \frac{6 x}{\cos (x) - 10 y} + \frac{y \sin (x)}{\cos (x) - 10 y}$

Simplify the right side.

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Combine the numerators over the common denominator.

$y' = \frac{6 x + y \sin (x)}{\cos (x) - 10 y}$

Step 6

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

$\frac{d y}{d x} = \frac{6 x + y \sin (x)}{\cos (x) - 10 y}$