Calculus Examples

$y = {(1 + 5 x)}^{10}$ , $(0, 1)$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d y} (y) = \frac{d}{d y} ({(1 + 5 x)}^{10})$

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 chain rule, which states that $\frac{d}{d y} [f (g (y))]$ is $f' (g (y)) g' (y)$ where $f (y) = y^{10}$ and $g (y) = 1 + 5 x$ .

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Step 3.1.1

To apply the Chain Rule, set $u$ as $1 + 5 x$ .

$\frac{d}{d u} [u^{10}] \frac{d}{d y} [1 + 5 x]$

Step 3.1.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = 10$ .

$10 u^{9} \frac{d}{d y} [1 + 5 x]$

Step 3.1.3

Replace all occurrences of $u$ with $1 + 5 x$ .

$10 {(1 + 5 x)}^{9} \frac{d}{d y} [1 + 5 x]$

Step 3.2

Differentiate.

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Step 3.2.1

By the Sum Rule, the derivative of $1 + 5 x$ with respect to $y$ is $\frac{d}{d y} [1] + \frac{d}{d y} [5 x]$ .

$10 {(1 + 5 x)}^{9} (\frac{d}{d y} [1] + \frac{d}{d y} [5 x])$

Step 3.2.2

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

$10 {(1 + 5 x)}^{9} (0 + \frac{d}{d y} [5 x])$

Step 3.2.3

Add $0$ and $\frac{d}{d y} [5 x]$ .

$10 {(1 + 5 x)}^{9} \frac{d}{d y} [5 x]$

Step 3.2.4

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

$10 {(1 + 5 x)}^{9} (5 \frac{d}{d y} [x])$

Step 3.2.5

Multiply $5$ by $10$ .

$50 {(1 + 5 x)}^{9} \frac{d}{d y} [x]$

Step 3.3

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

$50 {(1 + 5 x)}^{9} x'$

Step 4

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

$1 = 50 {(1 + 5 x)}^{9} x'$

Step 5

Solve for $x'$ .

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Step 5.1

Rewrite the equation as $50 {(1 + 5 x)}^{9} x' = 1$ .

$50 {(1 + 5 x)}^{9} x' = 1$

Step 5.2

Divide each term in $50 {(1 + 5 x)}^{9} x' = 1$ by $50 {(1 + 5 x)}^{9}$ and simplify.

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Step 5.2.1

Divide each term in $50 {(1 + 5 x)}^{9} x' = 1$ by $50 {(1 + 5 x)}^{9}$ .

$\frac{50 {(1 + 5 x)}^{9} x'}{50 {(1 + 5 x)}^{9}} = \frac{1}{50 {(1 + 5 x)}^{9}}$

Step 5.2.2

Simplify the left side.

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Step 5.2.2.1

Cancel the common factor of $50$ .

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Step 5.2.2.1.1

Cancel the common factor.

$\frac{50 {(1 + 5 x)}^{9} x'}{50 {(1 + 5 x)}^{9}} = \frac{1}{50 {(1 + 5 x)}^{9}}$

Step 5.2.2.1.2

Rewrite the expression.

$\frac{{(1 + 5 x)}^{9} x'}{{(1 + 5 x)}^{9}} = \frac{1}{50 {(1 + 5 x)}^{9}}$

Step 5.2.2.2

Cancel the common factor of ${(1 + 5 x)}^{9}$ .

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Step 5.2.2.2.1

Cancel the common factor.

$\frac{{(1 + 5 x)}^{9} x'}{{(1 + 5 x)}^{9}} = \frac{1}{50 {(1 + 5 x)}^{9}}$

Step 5.2.2.2.2

Divide $x'$ by $1$ .

$x' = \frac{1}{50 {(1 + 5 x)}^{9}}$

Step 6

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

$\frac{d x}{d y} = \frac{1}{50 {(1 + 5 x)}^{9}}$

Step 7

Replace $x$ with $0$ and $y$ with $1$ in the expression.

$\frac{d x}{d y} = \frac{1}{50 {(1 + 5 (0))}^{9}}$

Step 8

Simplify the result.

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Step 8.1

Simplify the denominator.

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Step 8.1.1

Multiply $5$ by $0$ .

$\frac{1}{50 {(1 + 0)}^{9}}$

Step 8.1.2

Add $1$ and $0$ .

$\frac{1}{50 \cdot 1^{9}}$

Step 8.1.3

One to any power is one.

$\frac{1}{50 \cdot 1}$

Step 8.2

Multiply $50$ by $1$ .

$\frac{1}{50}$