Find dy/dx y=e^(3x)sin(2x)

$y = e^{3 x} \sin (2 x)$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (y) = \frac{d}{d x} (e^{3 x} \sin (2 x))$

Step 2

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

$y'$

Step 3

Differentiate the right side of the equation.

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 $f (x) = e^{3 x}$ and $g (x) = \sin (2 x)$ .

$e^{3 x} \frac{d}{d x} [\sin (2 x)] + \sin (2 x) \frac{d}{d x} [e^{3 x}]$

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 $g (x) = 2 x$ .

To apply the Chain Rule, set $u_{1}$ as $2 x$ .

$e^{3 x} (\frac{d}{d u_{1}} [\sin (u_{1})] \frac{d}{d x} [2 x]) + \sin (2 x) \frac{d}{d x} [e^{3 x}]$

The derivative of $\sin (u_{1})$ with respect to $u_{1}$ is $\cos (u_{1})$ .

$e^{3 x} (\cos (u_{1}) \frac{d}{d x} [2 x]) + \sin (2 x) \frac{d}{d x} [e^{3 x}]$

Replace all occurrences of $u_{1}$ with $2 x$ .

$e^{3 x} (\cos (2 x) \frac{d}{d x} [2 x]) + \sin (2 x) \frac{d}{d x} [e^{3 x}]$

Differentiate.

Since $2$ is constant with respect to $x$ , the derivative of $2 x$ with respect to $x$ is $2 \frac{d}{d x} [x]$ .

$e^{3 x} (\cos (2 x) (2 \frac{d}{d x} [x])) + \sin (2 x) \frac{d}{d x} [e^{3 x}]$

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

$e^{3 x} (\cos (2 x) (2 \cdot 1)) + \sin (2 x) \frac{d}{d x} [e^{3 x}]$

Simplify the expression.

Multiply $2$ by $1$ .

$e^{3 x} (\cos (2 x) \cdot 2) + \sin (2 x) \frac{d}{d x} [e^{3 x}]$

Move $2$ to the left of $\cos (2 x)$ .

$e^{3 x} (2 \cdot \cos (2 x)) + \sin (2 x) \frac{d}{d x} [e^{3 x}]$

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 $g (x) = 3 x$ .

To apply the Chain Rule, set $u_{2}$ as $3 x$ .

$e^{3 x} (2 \cos (2 x)) + \sin (2 x) (\frac{d}{d u_{2}} [e^{u_{2}}] \frac{d}{d x} [3 x])$

Differentiate using the Exponential Rule which states that $\frac{d}{d u_{2}} [a^{u_{2}}]$ is $a^{u_{2}} \ln (a)$ where $a$ = $e$ .

$e^{3 x} (2 \cos (2 x)) + \sin (2 x) (e^{u_{2}} \frac{d}{d x} [3 x])$

Replace all occurrences of $u_{2}$ with $3 x$ .

$e^{3 x} (2 \cos (2 x)) + \sin (2 x) (e^{3 x} \frac{d}{d x} [3 x])$

Differentiate.

Since $3$ is constant with respect to $x$ , the derivative of $3 x$ with respect to $x$ is $3 \frac{d}{d x} [x]$ .

$e^{3 x} (2 \cos (2 x)) + \sin (2 x) (e^{3 x} (3 \frac{d}{d x} [x]))$

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

$e^{3 x} (2 \cos (2 x)) + \sin (2 x) (e^{3 x} (3 \cdot 1))$

Simplify the expression.

Multiply $3$ by $1$ .

$e^{3 x} (2 \cos (2 x)) + \sin (2 x) (e^{3 x} \cdot 3)$

Move $3$ to the left of $e^{3 x}$ .

$e^{3 x} (2 \cos (2 x)) + \sin (2 x) (3 \cdot e^{3 x})$

Reorder terms.

$2 e^{3 x} \cos (2 x) + 3 e^{3 x} \sin (2 x)$

Step 4

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

$y' = 2 e^{3 x} \cos (2 x) + 3 e^{3 x} \sin (2 x)$

Step 5

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

$\frac{d y}{d x} = 2 e^{3 x} \cos (2 x) + 3 e^{3 x} \sin (2 x)$

Find dy/dx y=e^(3x)sin(2x)

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