Calculus Examples

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

Step 1

Differentiate both sides of the equation.

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

Step 2

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

$y'$

Step 3

Differentiate the right side of the equation.

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Step 3.1

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

$e^{a x} \frac{d}{d b} [\sin (b x)]$

Step 3.2

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

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Step 3.2.1

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

$e^{a x} (\frac{d}{d u} [\sin (u)] \frac{d}{d b} [b x])$

Step 3.2.2

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

$e^{a x} (\cos (u) \frac{d}{d b} [b x])$

Step 3.2.3

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

$e^{a x} (\cos (b x) \frac{d}{d b} [b x])$

Step 3.3

Differentiate.

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Step 3.3.1

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

$e^{a x} (\cos (b x) (x \frac{d}{d b} [b]))$

Step 3.3.2

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

$e^{a x} (\cos (b x) (x \cdot 1))$

Step 3.3.3

Multiply $x$ by $1$ .

$e^{a x} (\cos (b x) x)$

Step 3.4

Simplify.

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Step 3.4.1

Reorder the factors of $e^{a x} (\cos (b x) x)$ .

$e^{a x} x \cos (b x)$

Step 3.4.2

Reorder factors in $e^{a x} x \cos (b x)$ .

$x e^{a x} \cos (b x)$

Step 4

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

$y' = x e^{a x} \cos (b x)$

Step 5

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

$\frac{d y}{d b} = x e^{a x} \cos (b x)$