Calculus Examples

y = sin (x y)

$y = \sin (x y)$

Step 1

Differentiate both sides of the equation.

\frac{d}{d x} (y) = \frac{d}{d x} (sin (x y))

$\frac{d}{d x} (y) = \frac{d}{d x} (\sin (x y))$

Step 2

The derivative of

y

$y$ with respect to

x

$x$ is

y'

$y'$ .

y'

$y'$

Step 3

Differentiate the right side of the equation.

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Step 3.1

Differentiate using the chain rule, which states that

\frac{d}{d x} [f (g (x))]

$\frac{d}{d x} [f (g (x))]$ is

f' (g (x)) g' (x)

$f' (g (x)) g' (x)$ where

f (x) = sin (x)

$f (x) = \sin (x)$ and

g (x) = x y

$g (x) = x y$ .

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Step 3.1.1

To apply the Chain Rule, set

u

$u$ as

x y

$x y$ .

\frac{d}{d u} [sin (u)] \frac{d}{d x} [x y]

$\frac{d}{d u} [\sin (u)] \frac{d}{d x} [x y]$

Step 3.1.2

The derivative of

sin (u)

$\sin (u)$ with respect to

u

$u$ is

cos (u)

$\cos (u)$ .

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

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

Step 3.1.3

Replace all occurrences of

u

$u$ with

x y

$x y$ .

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

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

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

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

Step 3.2

Differentiate using the Product Rule which states that

\frac{d}{d x} [f (x) g (x)]

$\frac{d}{d x} [f (x) g (x)]$ is

f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]

$f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where

f (x) = x

$f (x) = x$ and

g (x) = y

$g (x) = y$ .

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

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

Step 3.3

Rewrite

\frac{d}{d x} [y]

$\frac{d}{d x} [y]$ as

y'

$y'$ .

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

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

Step 3.4

Differentiate using the Power Rule which states that

\frac{d}{d x} [x^{n}]

$\frac{d}{d x} [x^{n}]$ is

n x^{n - 1}

$n x^{n - 1}$ where

n = 1

$n = 1$ .

cos (x y) (x y' + y \cdot 1)

$\cos (x y) (x y' + y \cdot 1)$

Step 3.5

Multiply

y

$y$ by

1

$1$ .

cos (x y) (x y' + y)

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

Step 3.6

Simplify.

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Step 3.6.1

Apply the distributive property.

cos (x y) (x y') + cos (x y) y

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

Step 3.6.2

Reorder terms.

x cos (x y) y' + y cos (x y)

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

x cos (x y) y' + y cos (x y)

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

x cos (x y) y' + y cos (x y)

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

Step 4

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

y' = x cos (x y) y' + y cos (x y)

$y' = x \cos (x y) y' + y \cos (x y)$

Step 5

Solve for

y'

$y'$ .

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Step 5.1

Simplify the right side.

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Step 5.1.1

Reorder factors in

x cos (x y) y' + y cos (x y)

$x \cos (x y) y' + y \cos (x y)$ .

y' = x y' cos (x y) + y cos (x y)

$y' = x y' \cos (x y) + y \cos (x y)$

y' = x y' cos (x y) + y cos (x y)

$y' = x y' \cos (x y) + y \cos (x y)$

Step 5.2

Subtract

x y' cos (x y)

$x y' \cos (x y)$ from both sides of the equation.

y' - x y' cos (x y) = y cos (x y)

$y' - x y' \cos (x y) = y \cos (x y)$

Step 5.3

Factor

y'

$y'$ out of

y' - x y' cos (x y)

$y' - x y' \cos (x y)$ .

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Step 5.3.1

Factor

y'

$y'$ out of

{y'}^{1}

${y'}^{1}$ .

y' \cdot 1 - x y' cos (x y) = y cos (x y)

$y' \cdot 1 - x y' \cos (x y) = y \cos (x y)$

Step 5.3.2

Factor

y'

$y'$ out of

- x y' cos (x y)

$- x y' \cos (x y)$ .

y' \cdot 1 + y' (- x cos (x y)) = y cos (x y)

$y' \cdot 1 + y' (- x \cos (x y)) = y \cos (x y)$

Step 5.3.3

Factor

y'

$y'$ out of

y' \cdot 1 + y' (- x cos (x y))

$y' \cdot 1 + y' (- x \cos (x y))$ .

y' (1 - x cos (x y)) = y cos (x y)

$y' (1 - x \cos (x y)) = y \cos (x y)$

y' (1 - x cos (x y)) = y cos (x y)

$y' (1 - x \cos (x y)) = y \cos (x y)$

Step 5.4

Divide each term in

y' (1 - x cos (x y)) = y cos (x y)

$y' (1 - x \cos (x y)) = y \cos (x y)$ by

1 - x cos (x y)

$1 - x \cos (x y)$ and simplify.

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Step 5.4.1

Divide each term in

y' (1 - x cos (x y)) = y cos (x y)

$y' (1 - x \cos (x y)) = y \cos (x y)$ by

1 - x cos (x y)

$1 - x \cos (x y)$ .

\frac{y' (1 - x cos (x y))}{1 - x cos (x y)} = \frac{y cos (x y)}{1 - x cos (x y)}

$\frac{y' (1 - x \cos (x y))}{1 - x \cos (x y)} = \frac{y \cos (x y)}{1 - x \cos (x y)}$

Step 5.4.2

Simplify the left side.

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Step 5.4.2.1

Cancel the common factor of

1 - x cos (x y)

$1 - x \cos (x y)$ .

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Step 5.4.2.1.1

Cancel the common factor.

$\frac{y' (1 - x \cos (x y))}{1 - x \cos (x y)} = \frac{y \cos (x y)}{1 - x \cos (x y)}$

Step 5.4.2.1.2

Divide

$y'$ by

$1$ .

$y' = \frac{y \cos (x y)}{1 - x \cos (x y)}$

Step 6

Replace

$y'$ with

$\frac{d y}{d x}$ .

$\frac{d y}{d x} = \frac{y \cos (x y)}{1 - x \cos (x y)}$