Calculus Examples

${(x + y^{2})}^{10} = 3 x^{2} + 7$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} ({(x + y^{2})}^{10}) = \frac{d}{d x} (3 x^{2} + 7)$

Step 2

Differentiate the left side of the equation.

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Step 2.1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{10}$ and $g (x) = x + y^{2}$ .

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Step 2.1.1

To apply the Chain Rule, set $u_{1}$ as $x + y^{2}$ .

$\frac{d}{d u_{1}} [{u_{1}}^{10}] \frac{d}{d x} [x + y^{2}]$

Step 2.1.2

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

$10 {u_{1}}^{9} \frac{d}{d x} [x + y^{2}]$

Step 2.1.3

Replace all occurrences of $u_{1}$ with $x + y^{2}$ .

$10 {(x + y^{2})}^{9} \frac{d}{d x} [x + y^{2}]$

Step 2.2

Differentiate.

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Step 2.2.1

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

$10 {(x + y^{2})}^{9} (\frac{d}{d x} [x] + \frac{d}{d x} [y^{2}])$

Step 2.2.2

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

$10 {(x + y^{2})}^{9} (1 + \frac{d}{d x} [y^{2}])$

Step 2.3

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 $g (x) = y$ .

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Step 2.3.1

To apply the Chain Rule, set $u_{2}$ as $y$ .

$10 {(x + y^{2})}^{9} (1 + \frac{d}{d u_{2}} [{u_{2}}^{2}] \frac{d}{d x} [y])$

Step 2.3.2

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

$10 {(x + y^{2})}^{9} (1 + 2 u_{2} \frac{d}{d x} [y])$

Step 2.3.3

Replace all occurrences of $u_{2}$ with $y$ .

$10 {(x + y^{2})}^{9} (1 + 2 y \frac{d}{d x} [y])$

Step 2.4

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

$10 {(x + y^{2})}^{9} (1 + 2 y y')$

Step 3

Differentiate the right side of the equation.

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Step 3.1

By the Sum Rule, the derivative of $3 x^{2} + 7$ with respect to $x$ is $\frac{d}{d x} [3 x^{2}] + \frac{d}{d x} [7]$ .

$\frac{d}{d x} [3 x^{2}] + \frac{d}{d x} [7]$

Step 3.2

Evaluate $\frac{d}{d x} [3 x^{2}]$ .

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Step 3.2.1

Since $3$ is constant with respect to $x$ , the derivative of $3 x^{2}$ with respect to $x$ is $3 \frac{d}{d x} [x^{2}]$ .

$3 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [7]$

Step 3.2.2

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

$3 (2 x) + \frac{d}{d x} [7]$

Step 3.2.3

Multiply $2$ by $3$ .

$6 x + \frac{d}{d x} [7]$

Step 3.3

Differentiate using the Constant Rule.

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Step 3.3.1

Since $7$ is constant with respect to $x$ , the derivative of $7$ with respect to $x$ is $0$ .

$6 x + 0$

Step 3.3.2

Add $6 x$ and $0$ .

$6 x$

Step 4

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

$10 {(x + y^{2})}^{9} (1 + 2 y y') = 6 x$

Step 5

Solve for $y'$ .

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Step 5.1

Divide each term in $10 {(x + y^{2})}^{9} (1 + 2 y y') = 6 x$ by $10 {(x + y^{2})}^{9}$ and simplify.

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Step 5.1.1

Divide each term in $10 {(x + y^{2})}^{9} (1 + 2 y y') = 6 x$ by $10 {(x + y^{2})}^{9}$ .

$\frac{10 {(x + y^{2})}^{9} (1 + 2 y y')}{10 {(x + y^{2})}^{9}} = \frac{6 x}{10 {(x + y^{2})}^{9}}$

Step 5.1.2

Simplify the left side.

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Step 5.1.2.1

Cancel the common factor of $10$ .

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Step 5.1.2.1.1

Cancel the common factor.

$\frac{10 {(x + y^{2})}^{9} (1 + 2 y y')}{10 {(x + y^{2})}^{9}} = \frac{6 x}{10 {(x + y^{2})}^{9}}$

Step 5.1.2.1.2

Rewrite the expression.

$\frac{{(x + y^{2})}^{9} (1 + 2 y y')}{{(x + y^{2})}^{9}} = \frac{6 x}{10 {(x + y^{2})}^{9}}$

Step 5.1.2.2

Cancel the common factor of ${(x + y^{2})}^{9}$ .

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Step 5.1.2.2.1

Cancel the common factor.

$\frac{{(x + y^{2})}^{9} (1 + 2 y y')}{{(x + y^{2})}^{9}} = \frac{6 x}{10 {(x + y^{2})}^{9}}$

Step 5.1.2.2.2

Divide $1 + 2 y y'$ by $1$ .

$1 + 2 y y' = \frac{6 x}{10 {(x + y^{2})}^{9}}$

Step 5.1.3

Simplify the right side.

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Step 5.1.3.1

Cancel the common factor of $6$ and $10$ .

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Step 5.1.3.1.1

Factor $2$ out of $6 x$ .

$1 + 2 y y' = \frac{2 (3 x)}{10 {(x + y^{2})}^{9}}$

Step 5.1.3.1.2

Cancel the common factors.

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Step 5.1.3.1.2.1

Factor $2$ out of $10 {(x + y^{2})}^{9}$ .

$1 + 2 y y' = \frac{2 (3 x)}{2 (5 {(x + y^{2})}^{9})}$

Step 5.1.3.1.2.2

Cancel the common factor.

$1 + 2 y y' = \frac{2 (3 x)}{2 (5 {(x + y^{2})}^{9})}$

Step 5.1.3.1.2.3

Rewrite the expression.

$1 + 2 y y' = \frac{3 x}{5 {(x + y^{2})}^{9}}$

Step 5.2

Subtract $1$ from both sides of the equation.

$2 y y' = \frac{3 x}{5 {(x + y^{2})}^{9}} - 1$

Step 5.3

Divide each term in $2 y y' = \frac{3 x}{5 {(x + y^{2})}^{9}} - 1$ by $2 y$ and simplify.

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Step 5.3.1

Divide each term in $2 y y' = \frac{3 x}{5 {(x + y^{2})}^{9}} - 1$ by $2 y$ .

$\frac{2 y y'}{2 y} = \frac{\frac{3 x}{5 {(x + y^{2})}^{9}}}{2 y} + \frac{- 1}{2 y}$

Step 5.3.2

Simplify the left side.

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Step 5.3.2.1

Cancel the common factor of $2$ .

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Step 5.3.2.1.1

Cancel the common factor.

$\frac{2 y y'}{2 y} = \frac{\frac{3 x}{5 {(x + y^{2})}^{9}}}{2 y} + \frac{- 1}{2 y}$

Step 5.3.2.1.2

Rewrite the expression.

$\frac{y y'}{y} = \frac{\frac{3 x}{5 {(x + y^{2})}^{9}}}{2 y} + \frac{- 1}{2 y}$

Step 5.3.2.2

Cancel the common factor of $y$ .

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Step 5.3.2.2.1

Cancel the common factor.

$\frac{y y'}{y} = \frac{\frac{3 x}{5 {(x + y^{2})}^{9}}}{2 y} + \frac{- 1}{2 y}$

Step 5.3.2.2.2

Divide $y'$ by $1$ .

$y' = \frac{\frac{3 x}{5 {(x + y^{2})}^{9}}}{2 y} + \frac{- 1}{2 y}$

Step 5.3.3

Simplify the right side.

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Step 5.3.3.1

Combine the numerators over the common denominator.

$y' = \frac{\frac{3 x}{5 {(x + y^{2})}^{9}} - 1}{2 y}$

Step 5.3.3.2

Simplify the numerator.

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Step 5.3.3.2.1

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{5 {(x + y^{2})}^{9}}{5 {(x + y^{2})}^{9}}$ .

$y' = \frac{\frac{3 x}{5 {(x + y^{2})}^{9}} - 1 \cdot \frac{5 {(x + y^{2})}^{9}}{5 {(x + y^{2})}^{9}}}{2 y}$

Step 5.3.3.2.2

Combine $- 1$ and $\frac{5 {(x + y^{2})}^{9}}{5 {(x + y^{2})}^{9}}$ .

$y' = \frac{\frac{3 x}{5 {(x + y^{2})}^{9}} + \frac{- (5 {(x + y^{2})}^{9})}{5 {(x + y^{2})}^{9}}}{2 y}$

Step 5.3.3.2.3

Combine the numerators over the common denominator.

$y' = \frac{\frac{3 x - (5 {(x + y^{2})}^{9})}{5 {(x + y^{2})}^{9}}}{2 y}$

Step 5.3.3.2.4

Multiply $- 1$ by $5$ .

$y' = \frac{\frac{3 x - 5 {(x + y^{2})}^{9}}{5 {(x + y^{2})}^{9}}}{2 y}$

Step 5.3.3.3

Multiply the numerator by the reciprocal of the denominator.

$y' = \frac{3 x - 5 {(x + y^{2})}^{9}}{5 {(x + y^{2})}^{9}} \cdot \frac{1}{2 y}$

Step 5.3.3.4

Multiply $\frac{3 x - 5 {(x + y^{2})}^{9}}{5 {(x + y^{2})}^{9}} \cdot \frac{1}{2 y}$ .

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Step 5.3.3.4.1

Multiply $\frac{3 x - 5 {(x + y^{2})}^{9}}{5 {(x + y^{2})}^{9}}$ by $\frac{1}{2 y}$ .

$y' = \frac{3 x - 5 {(x + y^{2})}^{9}}{5 {(x + y^{2})}^{9} (2 y)}$

Step 5.3.3.4.2

Multiply $2$ by $5$ .

$y' = \frac{3 x - 5 {(x + y^{2})}^{9}}{10 {(x + y^{2})}^{9} y}$

Step 5.3.3.5

Reorder factors in $\frac{3 x - 5 {(x + y^{2})}^{9}}{10 {(x + y^{2})}^{9} y}$ .

$y' = \frac{3 x - 5 {(x + y^{2})}^{9}}{10 y {(x + y^{2})}^{9}}$

Step 6

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

$\frac{d y}{d x} = \frac{3 x - 5 {(x + y^{2})}^{9}}{10 y {(x + y^{2})}^{9}}$