Calculus Examples

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

Step 1

Differentiate both sides of the equation.

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

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 chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{2}$ and $g (x) = x^{3} + 1$ .

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Step 3.1.1

To apply the Chain Rule, set $u$ as $x^{3} + 1$ .

$\frac{d}{d u} [u^{2}] \frac{d}{d x} [x^{3} + 1]$

Step 3.1.2

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

$2 u \frac{d}{d x} [x^{3} + 1]$

Step 3.1.3

Replace all occurrences of $u$ with $x^{3} + 1$ .

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

Step 3.2

Differentiate.

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Step 3.2.1

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

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

Step 3.2.2

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

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

Step 3.2.3

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

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

Step 3.2.4

Simplify the expression.

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Step 3.2.4.1

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

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

Step 3.2.4.2

Multiply $3$ by $2$ .

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

Step 3.3

Simplify.

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Step 3.3.1

Apply the distributive property.

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

Step 3.3.2

Apply the distributive property.

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

Step 3.3.3

Combine terms.

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Step 3.3.3.1

Multiply $x^{3}$ by $x^{2}$ by adding the exponents.

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Step 3.3.3.1.1

Move $x^{2}$ .

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

Step 3.3.3.1.2

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

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

Step 3.3.3.1.3

Add $2$ and $3$ .

$6 x^{5} + 6 \cdot 1 x^{2}$

Step 3.3.3.2

Multiply $6$ by $1$ .

$6 x^{5} + 6 x^{2}$

Step 4

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

$y' = 6 x^{5} + 6 x^{2}$

Step 5

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

$\frac{d y}{d x} = 6 x^{5} + 6 x^{2}$