Calculus Examples

$y - x \sin (x y)$

Step 1

Differentiate.

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Step 1.1

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

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

Step 1.2

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

$0 + \frac{d}{d x} [- x \sin (x y)]$

Step 2

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

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Step 2.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- x \sin (x y)$ with respect to $x$ is $- \frac{d}{d x} [x \sin (x y)]$ .

$0 - \frac{d}{d x} [x \sin (x y)]$

Step 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 $f (x) = x$ and $g (x) = \sin (x y)$ .

$0 - (x \frac{d}{d x} [\sin (x y)] + \sin (x y) \frac{d}{d x} [x])$

Step 2.3

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

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Step 2.3.1

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

$0 - (x (\frac{d}{d u} [\sin (u)] \frac{d}{d x} [x y]) + \sin (x y) \frac{d}{d x} [x])$

Step 2.3.2

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

$0 - (x (\cos (u) \frac{d}{d x} [x y]) + \sin (x y) \frac{d}{d x} [x])$

Step 2.3.3

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

$0 - (x (\cos (x y) \frac{d}{d x} [x y]) + \sin (x y) \frac{d}{d x} [x])$

Step 2.4

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

$0 - (x (\cos (x y) (y \frac{d}{d x} [x])) + \sin (x y) \frac{d}{d x} [x])$

Step 2.5

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

$0 - (x (\cos (x y) (y \cdot 1)) + \sin (x y) \frac{d}{d x} [x])$

Step 2.6

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

$0 - (x (\cos (x y) (y \cdot 1)) + \sin (x y) \cdot 1)$

Step 2.7

Multiply $y$ by $1$ .

$0 - (x (\cos (x y) y) + \sin (x y) \cdot 1)$

Step 2.8

Multiply $\sin (x y)$ by $1$ .

$0 - (x (\cos (x y) y) + \sin (x y))$

Step 3

Simplify.

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Step 3.1

Apply the distributive property.

$0 - (x (\cos (x y) y)) - \sin (x y)$

Step 3.2

Subtract $- (- x (\cos (x y) y) - \sin (x y))$ from $0$ .

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

Step 3.3

Reorder terms.

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