Calculus Examples

$y = (x^{3} + 4 + 5 x^{2}) \sin (x)$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (y) = \frac{d}{d x} ((x^{3} + 4 + 5 x^{2}) \sin (x))$

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 $f (x) = x^{3} + 4 + 5 x^{2}$ and $g (x) = \sin (x)$ .

$(x^{3} + 4 + 5 x^{2}) \frac{d}{d x} [\sin (x)] + \sin (x) \frac{d}{d x} [x^{3} + 4 + 5 x^{2}]$

Step 3.2

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

$(x^{3} + 4 + 5 x^{2}) \cos (x) + \sin (x) \frac{d}{d x} [x^{3} + 4 + 5 x^{2}]$

Step 3.3

Differentiate.

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Step 3.3.1

By the Sum Rule, the derivative of $x^{3} + 4 + 5 x^{2}$ with respect to $x$ is $\frac{d}{d x} [x^{3}] + \frac{d}{d x} [4] + \frac{d}{d x} [5 x^{2}]$ .

$(x^{3} + 4 + 5 x^{2}) \cos (x) + \sin (x) (\frac{d}{d x} [x^{3}] + \frac{d}{d x} [4] + \frac{d}{d x} [5 x^{2}])$

Step 3.3.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

$(x^{3} + 4 + 5 x^{2}) \cos (x) + \sin (x) (3 x^{2} + \frac{d}{d x} [4] + \frac{d}{d x} [5 x^{2}])$

Step 3.3.3

Since $4$ is constant with respect to $x$ , the derivative of $4$ with respect to $x$ is $0$ .

$(x^{3} + 4 + 5 x^{2}) \cos (x) + \sin (x) (3 x^{2} + 0 + \frac{d}{d x} [5 x^{2}])$

Step 3.3.4

Add $3 x^{2}$ and $0$ .

$(x^{3} + 4 + 5 x^{2}) \cos (x) + \sin (x) (3 x^{2} + \frac{d}{d x} [5 x^{2}])$

Step 3.3.5

Since $5$ is constant with respect to $x$ , the derivative of $5 x^{2}$ with respect to $x$ is $5 \frac{d}{d x} [x^{2}]$ .

$(x^{3} + 4 + 5 x^{2}) \cos (x) + \sin (x) (3 x^{2} + 5 \frac{d}{d x} [x^{2}])$

Step 3.3.6

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

$(x^{3} + 4 + 5 x^{2}) \cos (x) + \sin (x) (3 x^{2} + 5 (2 x))$

Step 3.3.7

Multiply $2$ by $5$ .

$(x^{3} + 4 + 5 x^{2}) \cos (x) + \sin (x) (3 x^{2} + 10 x)$

Step 3.4

Simplify.

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Step 3.4.1

Apply the distributive property.

$x^{3} \cos (x) + 4 \cos (x) + 5 x^{2} \cos (x) + \sin (x) (3 x^{2} + 10 x)$

Step 3.4.2

Apply the distributive property.

$x^{3} \cos (x) + 4 \cos (x) + 5 x^{2} \cos (x) + \sin (x) (3 x^{2}) + \sin (x) (10 x)$

Step 3.4.3

Reorder terms.

$x^{3} \cos (x) + 5 x^{2} \cos (x) + 3 x^{2} \sin (x) + 10 x \sin (x) + 4 \cos (x)$

Step 4

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

$y' = x^{3} \cos (x) + 5 x^{2} \cos (x) + 3 x^{2} \sin (x) + 10 x \sin (x) + 4 \cos (x)$

Step 5

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

$\frac{d y}{d x} = x^{3} \cos (x) + 5 x^{2} \cos (x) + 3 x^{2} \sin (x) + 10 x \sin (x) + 4 \cos (x)$