Calculus Examples

$y = (x^{2} + 5) (3 x + 1)$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (y) = \frac{d}{d x} ((x^{2} + 5) (3 x + 1))$

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 $f (x) = x^{2} + 5$ and $g (x) = 3 x + 1$ .

$(x^{2} + 5) \frac{d}{d x} [3 x + 1] + (3 x + 1) \frac{d}{d x} [x^{2} + 5]$

Step 3.2

Differentiate.

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Step 3.2.1

By the Sum Rule, the derivative of $3 x + 1$ with respect to $x$ is $\frac{d}{d x} [3 x] + \frac{d}{d x} [1]$ .

$(x^{2} + 5) (\frac{d}{d x} [3 x] + \frac{d}{d x} [1]) + (3 x + 1) \frac{d}{d x} [x^{2} + 5]$

Step 3.2.2

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

$(x^{2} + 5) (3 \frac{d}{d x} [x] + \frac{d}{d x} [1]) + (3 x + 1) \frac{d}{d x} [x^{2} + 5]$

Step 3.2.3

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

$(x^{2} + 5) (3 \cdot 1 + \frac{d}{d x} [1]) + (3 x + 1) \frac{d}{d x} [x^{2} + 5]$

Step 3.2.4

Multiply $3$ by $1$ .

$(x^{2} + 5) (3 + \frac{d}{d x} [1]) + (3 x + 1) \frac{d}{d x} [x^{2} + 5]$

Step 3.2.5

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$(x^{2} + 5) (3 + 0) + (3 x + 1) \frac{d}{d x} [x^{2} + 5]$

Step 3.2.6

Simplify the expression.

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Step 3.2.6.1

Add $3$ and $0$ .

$(x^{2} + 5) \cdot 3 + (3 x + 1) \frac{d}{d x} [x^{2} + 5]$

Step 3.2.6.2

Move $3$ to the left of $x^{2} + 5$ .

$3 \cdot (x^{2} + 5) + (3 x + 1) \frac{d}{d x} [x^{2} + 5]$

Step 3.2.7

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

$3 (x^{2} + 5) + (3 x + 1) (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [5])$

Step 3.2.8

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

$3 (x^{2} + 5) + (3 x + 1) (2 x + \frac{d}{d x} [5])$

Step 3.2.9

Since $5$ is constant with respect to $x$ , the derivative of $5$ with respect to $x$ is $0$ .

$3 (x^{2} + 5) + (3 x + 1) (2 x + 0)$

Step 3.2.10

Simplify the expression.

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Step 3.2.10.1

Add $2 x$ and $0$ .

$3 (x^{2} + 5) + (3 x + 1) (2 x)$

Step 3.2.10.2

Move $2$ to the left of $3 x + 1$ .

$3 (x^{2} + 5) + 2 \cdot (3 x + 1) x$

Step 3.3

Simplify.

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Step 3.3.1

Apply the distributive property.

$3 x^{2} + 3 \cdot 5 + 2 (3 x + 1) x$

Step 3.3.2

Apply the distributive property.

$3 x^{2} + 3 \cdot 5 + (2 (3 x) + 2 \cdot 1) x$

Step 3.3.3

Apply the distributive property.

$3 x^{2} + 3 \cdot 5 + 2 (3 x) x + 2 \cdot 1 x$

Step 3.3.4

Combine terms.

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Step 3.3.4.1

Multiply $3$ by $5$ .

$3 x^{2} + 15 + 2 (3 x) x + 2 \cdot 1 x$

Step 3.3.4.2

Multiply $3$ by $2$ .

$3 x^{2} + 15 + 6 x \cdot x + 2 \cdot 1 x$

Step 3.3.4.3

Raise $x$ to the power of $1$ .

$3 x^{2} + 15 + 6 (x^{1} x) + 2 \cdot 1 x$

Step 3.3.4.4

Raise $x$ to the power of $1$ .

$3 x^{2} + 15 + 6 (x^{1} x^{1}) + 2 \cdot 1 x$

Step 3.3.4.5

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$3 x^{2} + 15 + 6 x^{1 + 1} + 2 \cdot 1 x$

Step 3.3.4.6

Add $1$ and $1$ .

$3 x^{2} + 15 + 6 x^{2} + 2 \cdot 1 x$

Step 3.3.4.7

Multiply $2$ by $1$ .

$3 x^{2} + 15 + 6 x^{2} + 2 x$

Step 3.3.4.8

Add $3 x^{2}$ and $6 x^{2}$ .

$9 x^{2} + 15 + 2 x$

Step 3.3.5

Reorder terms.

$9 x^{2} + 2 x + 15$

Step 4

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

$y' = 9 x^{2} + 2 x + 15$

Step 5

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

$\frac{d y}{d x} = 9 x^{2} + 2 x + 15$