Calculus Examples

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

Step 1

Differentiate both sides of the equation.

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

Step 2

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

$1$

Step 3

Differentiate the right side of the equation.

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Step 3.1

Differentiate using the Product Rule which states that $\frac{d}{d y} [f (y) g (y)]$ is $f (y) \frac{d}{d y} [g (y)] + g (y) \frac{d}{d y} [f (y)]$ where $f (y) = x^{2} + 5$ and $g (y) = 3 x + 1$ .

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

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 $y$ is $\frac{d}{d y} [3 x] + \frac{d}{d y} [1]$ .

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

Step 3.2.2

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

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

Step 3.3

Rewrite $\frac{d}{d y} [x]$ as $x'$ .

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

Step 3.4

Differentiate.

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Step 3.4.1

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

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

Step 3.4.2

Simplify the expression.

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Step 3.4.2.1

Add $3 x'$ and $0$ .

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

Step 3.4.2.2

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

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

Step 3.4.3

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

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

Step 3.5

Differentiate using the chain rule, which states that $\frac{d}{d y} [f (g (y))]$ is $f' (g (y)) g' (y)$ where $f (y) = y^{2}$ and $g (y) = x$ .

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Step 3.5.1

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

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

Step 3.5.2

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

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

Step 3.5.3

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

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

Step 3.6

Rewrite $\frac{d}{d y} [x]$ as $x'$ .

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

Step 3.7

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

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

Step 3.8

Simplify the expression.

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Step 3.8.1

Add $2 x x'$ and $0$ .

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

Step 3.8.2

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

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

Step 3.9

Simplify.

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Step 3.9.1