Calculus Examples

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

Step 1

Differentiate both sides of the equation.

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

Step 2

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

$y'$

Step 3

Differentiate the right side of the equation.

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Step 3.1

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 $f (x) = e^{- x}$ and $g (x) = \sin (x)$ .

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

Step 3.2

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

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

Step 3.3

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

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Step 3.3.1

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

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

Step 3.3.2

Differentiate using the Exponential Rule which states that $\frac{d}{d u} [a^{u}]$ is $a^{u} \ln (a)$ where $a$ = $e$ .

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

Step 3.3.3

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

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

Step 3.4

Differentiate.

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Step 3.4.1

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

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

Step 3.4.2

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

$e^{- x} \cos (x) + \sin (x) (e^{- x} (- 1 \cdot 1))$

Step 3.4.3

Simplify the expression.

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Step 3.4.3.1

Multiply $- 1$ by $1$ .

$e^{- x} \cos (x) + \sin (x) (e^{- x} \cdot - 1)$

Step 3.4.3.2

Move $- 1$ to the left of $e^{- x}$ .

$e^{- x} \cos (x) + \sin (x) (- 1 \cdot e^{- x})$

Step 3.4.3.3

Rewrite $- 1 e^{- x}$ as $- e^{- x}$ .

$e^{- x} \cos (x) + \sin (x) (- e^{- x})$

Step 3.4.3.4

Reorder terms.

$e^{- x} \cos (x) - e^{- x} \sin (x)$

Step 4

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

$y' = e^{- x} \cos (x) - e^{- x} \sin (x)$

Step 5

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

$\frac{d y}{d x} = e^{- x} \cos (x) - e^{- x} \sin (x)$