Calculus Examples

$y = x^{4} + 8$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d y} (y) = \frac{d}{d y} (x^{4} + 8)$

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

By the Sum Rule, the derivative of $x^{4} + 8$ with respect to $y$ is $\frac{d}{d y} [x^{4}] + \frac{d}{d y} [8]$ .

$\frac{d}{d y} [x^{4}] + \frac{d}{d y} [8]$

Step 3.2

Evaluate $\frac{d}{d y} [x^{4}]$ .

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Step 3.2.1

Differentiate using the chain rule, which states that $\frac{d}{d y} [f (g (y))]$ is $f' (g (y)) g' (y)$ where $f (y) = y^{4}$ and $g (y) = x$ .

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Step 3.2.1.1

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

$\frac{d}{d u} [u^{4}] \frac{d}{d y} [x] + \frac{d}{d y} [8]$

Step 3.2.1.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = 4$ .

$4 u^{3} \frac{d}{d y} [x] + \frac{d}{d y} [8]$

Step 3.2.1.3

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

$4 x^{3} \frac{d}{d y} [x] + \frac{d}{d y} [8]$

Step 3.2.2

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

$4 x^{3} x' + \frac{d}{d y} [8]$

Step 3.3

Differentiate using the Constant Rule.

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Step 3.3.1

Since $8$ is constant with respect to $y$ , the derivative of $8$ with respect to $y$ is $0$ .

$4 x^{3} x' + 0$

Step 3.3.2

Add $4 x^{3} x'$ and $0$ .

$4 x^{3} x'$

Step 4

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

$1 = 4 x^{3} x'$

Step 5

Solve for $x'$ .

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Step 5.1

Rewrite the equation as $4 x^{3} x' = 1$ .

$4 x^{3} x' = 1$

Step 5.2

Divide each term in $4 x^{3} x' = 1$ by $4 x^{3}$ and simplify.

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Step 5.2.1

Divide each term in $4 x^{3} x' = 1$ by $4 x^{3}$ .

$\frac{4 x^{3} x'}{4 x^{3}} = \frac{1}{4 x^{3}}$

Step 5.2.2

Simplify the left side.

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Step 5.2.2.1

Cancel the common factor of $4$ .

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Step 5.2.2.1.1

Cancel the common factor.

$\frac{4 x^{3} x'}{4 x^{3}} = \frac{1}{4 x^{3}}$

Step 5.2.2.1.2

Rewrite the expression.

$\frac{x^{3} x'}{x^{3}} = \frac{1}{4 x^{3}}$

Step 5.2.2.2

Cancel the common factor of $x^{3}$ .

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Step 5.2.2.2.1

Cancel the common factor.

$\frac{x^{3} x'}{x^{3}} = \frac{1}{4 x^{3}}$

Step 5.2.2.2.2

Divide $x'$ by $1$ .

$x' = \frac{1}{4 x^{3}}$

Step 6

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

$\frac{d x}{d y} = \frac{1}{4 x^{3}}$

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