Calculus Examples

$5 x^{2} y - π \cos (y) = 6 π$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (5 x^{2} y - π \cos (y)) = \frac{d}{d x} (6 π)$

Step 2

Differentiate the left side of the equation.

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Step 2.1

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

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

Step 2.2

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

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Step 2.2.1

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

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

Step 2.2.2

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 $f (x) = x^{2}$ and $g (x) = y$ .

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

Step 2.2.3

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

$5 (x^{2} y' + y \frac{d}{d x} [x^{2}]) + \frac{d}{d x} [- π \cos (y)]$

Step 2.2.4

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

$5 (x^{2} y' + y (2 x)) + \frac{d}{d x} [- π \cos (y)]$

Step 2.2.5

Move $2$ to the left of $y$ .

$5 (x^{2} y' + 2 y x) + \frac{d}{d x} [- π \cos (y)]$

Step 2.3

Evaluate $\frac{d}{d x} [- π \cos (y)]$ .

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Step 2.3.1

Since $- π$ is constant with respect to $x$ , the derivative of $- π \cos (y)$ with respect to $x$ is $- π \frac{d}{d x} [\cos (y)]$ .

$5 (x^{2} y' + 2 y x) - π \frac{d}{d x} [\cos (y)]$

Step 2.3.2

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

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Step 2.3.2.1

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

$5 (x^{2} y' + 2 y x) - π (\frac{d}{d u} [\cos (u)] \frac{d}{d x} [y])$

Step 2.3.2.2

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

$5 (x^{2} y' + 2 y x) - π (- \sin (u) \frac{d}{d x} [y])$

Step 2.3.2.3

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

$5 (x^{2} y' + 2 y x) - π (- \sin (y) \frac{d}{d x} [y])$

Step 2.3.3

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

$5 (x^{2} y' + 2 y x) - π (- \sin (y) y')$

Step 2.3.4

Multiply $- 1$ by $- 1$ .

$5 (x^{2} y' + 2 y x) + 1 π (\sin (y) y')$

Step 2.3.5

Multiply $π$ by $1$ .

$5 (x^{2} y' + 2 y x) + π \sin (y) y'$

Step 2.4

Simplify.

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Step 2.4.1

Apply the distributive property.

$5 (x^{2} y') + 5 (2 y x) + π \sin (y) y'$

Step 2.4.2

Multiply $2$ by $5$ .

$5 x^{2} y' + 10 y x + π \sin (y) y'$

Step 2.4.3

Reorder terms.

$5 x^{2} y' + 10 x y + π \sin (y) y'$

Step 3

Since $6 π$ is constant with respect to $x$ , the derivative of $6 π$ with respect to $x$ is $0$ .

$0$

Step 4

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

$5 x^{2} y' + 10 x y + π \sin (y) y' = 0$

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 $5 x^{2} y' + 10 x y + π \sin (y) y'$ .

$5 x^{2} y' + 10 x y + π y' \sin (y) = 0$

Step 5.2

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

$5 x^{2} y' + π y' \sin (y) = - 10 x y$

Step 5.3

Factor $y'$ out of $5 x^{2} y' + π y' \sin (y)$ .

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Step 5.3.1

Factor $y'$ out of $5 x^{2} y'$ .

$y' (5 x^{2}) + π y' \sin (y) = - 10 x y$

Step 5.3.2

Factor $y'$ out of $π y' \sin (y)$ .

$y' (5 x^{2}) + y' (π \sin (y)) = - 10 x y$

Step 5.3.3

Factor $y'$ out of $y' (5 x^{2}) + y' (π \sin (y))$ .

$y' (5 x^{2} + π \sin (y)) = - 10 x y$

Step 5.4

Divide each term in $y' (5 x^{2} + π \sin (y)) = - 10 x y$ by $5 x^{2} + π \sin (y)$ and simplify.

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Step 5.4.1

Divide each term in $y' (5 x^{2} + π \sin (y)) = - 10 x y$ by $5 x^{2} + π \sin (y)$ .

$\frac{y' (5 x^{2} + π \sin (y))}{5 x^{2} + π \sin (y)} = \frac{- 10 x y}{5 x^{2} + π \sin (y)}$

Step 5.4.2

Simplify the left side.

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Step 5.4.2.1

Cancel the common factor of $5 x^{2} + π \sin (y)$ .

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Step 5.4.2.1.1

Cancel the common factor.

$\frac{y' (5 x^{2} + π \sin (y))}{5 x^{2} + π \sin (y)} = \frac{- 10 x y}{5 x^{2} + π \sin (y)}$

Step 5.4.2.1.2

Divide $y'$ by $1$ .

$y' = \frac{- 10 x y}{5 x^{2} + π \sin (y)}$

Step 5.4.3

Simplify the right side.

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Step 5.4.3.1

Move the negative in front of the fraction.

$y' = - \frac{10 x y}{5 x^{2} + π \sin (y)}$

Step 6

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

$\frac{d y}{d x} = - \frac{10 x y}{5 x^{2} + π \sin (y)}$