Calculus Examples

$y = (4 x^{2} - e^{2 x}) \sin (3 x)$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (y) = \frac{d}{d x} ((4 x^{2} - e^{2 x}) \sin (3 x))$

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 $f (x) = 4 x^{2} - e^{2 x}$ and $g (x) = \sin (3 x)$ .

$(4 x^{2} - e^{2 x}) \frac{d}{d x} [\sin (3 x)] + \sin (3 x) \frac{d}{d x} [4 x^{2} - e^{2 x}]$

Step 3.2

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 $g (x) = 3 x$ .

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Step 3.2.1

To apply the Chain Rule, set $u_{1}$ as $3 x$ .

$(4 x^{2} - e^{2 x}) (\frac{d}{d u_{1}} [\sin (u_{1})] \frac{d}{d x} [3 x]) + \sin (3 x) \frac{d}{d x} [4 x^{2} - e^{2 x}]$

Step 3.2.2

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

$(4 x^{2} - e^{2 x}) (\cos (u_{1}) \frac{d}{d x} [3 x]) + \sin (3 x) \frac{d}{d x} [4 x^{2} - e^{2 x}]$

Step 3.2.3

Replace all occurrences of $u_{1}$ with $3 x$ .

$(4 x^{2} - e^{2 x}) (\cos (3 x) \frac{d}{d x} [3 x]) + \sin (3 x) \frac{d}{d x} [4 x^{2} - e^{2 x}]$

Step 3.3

Differentiate.

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Step 3.3.1

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

$(4 x^{2} - e^{2 x}) \cos (3 x) (3 \frac{d}{d x} [x]) + \sin (3 x) \frac{d}{d x} [4 x^{2} - e^{2 x}]$

Step 3.3.2

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

$(4 x^{2} - e^{2 x}) \cos (3 x) (3 \cdot 1) + \sin (3 x) \frac{d}{d x} [4 x^{2} - e^{2 x}]$

Step 3.3.3

Simplify the expression.

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Step 3.3.3.1

Multiply $3$ by $1$ .

$(4 x^{2} - e^{2 x}) \cos (3 x) \cdot 3 + \sin (3 x) \frac{d}{d x} [4 x^{2} - e^{2 x}]$

Step 3.3.3.2

Move $3$ to the left of $(4 x^{2} - e^{2 x}) \cos (3 x)$ .

$3 \cdot ((4 x^{2} - e^{2 x}) \cos (3 x)) + \sin (3 x) \frac{d}{d x} [4 x^{2} - e^{2 x}]$

Step 3.3.4

By the Sum Rule, the derivative of $4 x^{2} - e^{2 x}$ with respect to $x$ is $\frac{d}{d x} [4 x^{2}] + \frac{d}{d x} [- e^{2 x}]$ .

$3 (4 x^{2} - e^{2 x}) \cos (3 x) + \sin (3 x) (\frac{d}{d x} [4 x^{2}] + \frac{d}{d x} [- e^{2 x}])$

Step 3.3.5

Since $4$ is constant with respect to $x$ , the derivative of $4 x^{2}$ with respect to $x$ is $4 \frac{d}{d x} [x^{2}]$ .

$3 (4 x^{2} - e^{2 x}) \cos (3 x) + \sin (3 x) (4 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- e^{2 x}])$

Step 3.3.6

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$3 (4 x^{2} - e^{2 x}) \cos (3 x) + \sin (3 x) (4 (2 x) + \frac{d}{d x} [- e^{2 x}])$

Step 3.3.7

Multiply $2$ by $4$ .

$3 (4 x^{2} - e^{2 x}) \cos (3 x) + \sin (3 x) (8 x + \frac{d}{d x} [- e^{2 x}])$

Step 3.3.8

Since $- 1$ is constant with respect to $x$ , the derivative of $- e^{2 x}$ with respect to $x$ is $- \frac{d}{d x} [e^{2 x}]$ .

$3 (4 x^{2} - e^{2 x}) \cos (3 x) + \sin (3 x) (8 x - \frac{d}{d x} [e^{2 x}])$

Step 3.4

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 $g (x) = 2 x$ .

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Step 3.4.1

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

$3 (4 x^{2} - e^{2 x}) \cos (3 x) + \sin (3 x) (8 x - (\frac{d}{d u_{2}} [e^{u_{2}}] \frac{d}{d x} [2 x]))$

Step 3.4.2

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

$3 (4 x^{2} - e^{2 x}) \cos (3 x) + \sin (3 x) (8 x - (e^{u_{2}} \frac{d}{d x} [2 x]))$

Step 3.4.3

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

$3 (4 x^{2} - e^{2 x}) \cos (3 x) + \sin (3 x) (8 x - (e^{2 x} \frac{d}{d x} [2 x]))$

Step 3.5

Differentiate.

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Step 3.5.1

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

$3 (4 x^{2} - e^{2 x}) \cos (3 x) + \sin (3 x) (8 x - e^{2 x} (2 \frac{d}{d x} [x]))$

Step 3.5.2

Multiply $2$ by $- 1$ .

$3 (4 x^{2} - e^{2 x}) \cos (3 x) + \sin (3 x) (8 x - 2 e^{2 x} \frac{d}{d x} [x])$

Step 3.5.3

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

$3 (4 x^{2} - e^{2 x}) \cos (3 x) + \sin (3 x) (8 x - 2 e^{2 x} \cdot 1)$

Step 3.5.4

Multiply $- 2$ by $1$ .

$3 (4 x^{2} - e^{2 x}) \cos (3 x) + \sin (3 x) (8 x - 2 e^{2 x})$

Step 3.6

Simplify.

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Step 3.6.1

Apply the distributive property.

$(3 (4 x^{2}) + 3 (- e^{2 x})) \cos (3 x) + \sin (3 x) (8 x - 2 e^{2 x})$

Step 3.6.2

Apply the distributive property.

$3 (4 x^{2}) \cos (3 x) + 3 (- e^{2 x}) \cos (3 x) + \sin (3 x) (8 x - 2 e^{2 x})$

Step 3.6.3

Apply the distributive property.

$3 (4 x^{2}) \cos (3 x) + 3 (- e^{2 x}) \cos (3 x) + \sin (3 x) (8 x) + \sin (3 x) (- 2 e^{2 x})$

Step 3.6.4

Combine terms.

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Step 3.6.4.1

Multiply $4$ by $3$ .

$12 x^{2} \cos (3 x) + 3 (- e^{2 x}) \cos (3 x) + \sin (3 x) (8 x) + \sin (3 x) (- 2 e^{2 x})$

Step 3.6.4.2

Multiply $- 1$ by $3$ .

$12 x^{2} \cos (3 x) - 3 e^{2 x} \cos (3 x) + \sin (3 x) (8 x) + \sin (3 x) (- 2 e^{2 x})$

Step 3.6.5

Reorder terms.

$12 x^{2} \cos (3 x) - 3 e^{2 x} \cos (3 x) + 8 x \sin (3 x) - 2 e^{2 x} \sin (3 x)$

Step 4

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

$y' = 12 x^{2} \cos (3 x) - 3 e^{2 x} \cos (3 x) + 8 x \sin (3 x) - 2 e^{2 x} \sin (3 x)$

Step 5

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

$\frac{d y}{d x} = 12 x^{2} \cos (3 x) - 3 e^{2 x} \cos (3 x) + 8 x \sin (3 x) - 2 e^{2 x} \sin (3 x)$