Calculus Examples

$f = e^{x} \sin (y + z) - z$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d y} (f) = \frac{d}{d y} (e^{x} \sin (y + z) - z)$

Step 2

The derivative of $f$ with respect to $y$ is $f'$ .

$f'$

Step 3

Differentiate the right side of the equation.

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Step 3.1

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

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

Step 3.2

Evaluate $\frac{d}{d y} [e^{x} \sin (y + z)]$ .

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Step 3.2.1

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

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

Step 3.2.2

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

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Step 3.2.2.1

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

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

Step 3.2.2.2

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

$e^{x} (\cos (u) \frac{d}{d y} [y + z]) + \frac{d}{d y} [- z]$

Step 3.2.2.3

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

$e^{x} (\cos (y + z) \frac{d}{d y} [y + z]) + \frac{d}{d y} [- z]$

Step 3.2.3

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

$e^{x} (\cos (y + z) (\frac{d}{d y} [y] + \frac{d}{d y} [z])) + \frac{d}{d y} [- z]$

Step 3.2.4

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

$e^{x} (\cos (y + z) (1 + \frac{d}{d y} [z])) + \frac{d}{d y} [- z]$

Step 3.2.5

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

$e^{x} (\cos (y + z) (1 + 0)) + \frac{d}{d y} [- z]$

Step 3.2.6

Add $1$ and $0$ .

$e^{x} (\cos (y + z) \cdot 1) + \frac{d}{d y} [- z]$

Step 3.2.7

Multiply $\cos (y + z)$ by $1$ .

$e^{x} \cos (y + z) + \frac{d}{d y} [- z]$

Step 3.3

Differentiate using the Constant Rule.

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Step 3.3.1

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

$e^{x} \cos (y + z) + 0$

Step 3.3.2

Add $e^{x} \cos (y + z)$ and $0$ .

$e^{x} \cos (y + z)$

Step 4

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

$f' = e^{x} \cos (y + z)$

Step 5

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

$\frac{d f}{d y} = e^{x} \cos (y + z)$