Calculus Examples

$\tan (x^{2} y^{4}) = 3 x + y^{2}$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (\tan (x^{2} y^{4})) = \frac{d}{d x} (3 x + y^{2})$

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 $f (x) = \tan (x)$ and $g (x) = x^{2} y^{4}$ .

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Step 2.1.1

To apply the Chain Rule, set $u_{1}$ as $x^{2} y^{4}$ .

$\frac{d}{d u_{1}} [\tan (u_{1})] \frac{d}{d x} [x^{2} y^{4}]$

Step 2.1.2

The derivative of $\tan (u_{1})$ with respect to $u_{1}$ is $\sec^{2} (u_{1})$ .

$\sec^{2} (u_{1}) \frac{d}{d x} [x^{2} y^{4}]$

Step 2.1.3

Replace all occurrences of $u_{1}$ with $x^{2} y^{4}$ .

$\sec^{2} (x^{2} y^{4}) \frac{d}{d x} [x^{2} y^{4}]$

Step 2.2

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = x^{2}$ and $g (x) = y^{4}$ .

$\sec^{2} (x^{2} y^{4}) (x^{2} \frac{d}{d x} [y^{4}] + y^{4} \frac{d}{d x} [x^{2}])$

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 $f (x) = x^{4}$ and $g (x) = y$ .

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Step 2.3.1

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

$\sec^{2} (x^{2} y^{4}) (x^{2} (\frac{d}{d u_{2}} [{u_{2}}^{4}] \frac{d}{d x} [y]) + y^{4} \frac{d}{d x} [x^{2}])$

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 $n = 4$ .

$\sec^{2} (x^{2} y^{4}) (x^{2} (4 {u_{2}}^{3} \frac{d}{d x} [y]) + y^{4} \frac{d}{d x} [x^{2}])$

Step 2.3.3

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

$\sec^{2} (x^{2} y^{4}) (x^{2} (4 y^{3} \frac{d}{d x} [y]) + y^{4} \frac{d}{d x} [x^{2}])$

Step 2.4

Move $4$ to the left of $x^{2}$ .

$\sec^{2} (x^{2} y^{4}) (4 \cdot x^{2} y^{3} \frac{d}{d x} [y] + y^{4} \frac{d}{d x} [x^{2}])$

Step 2.5

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

$\sec^{2} (x^{2} y^{4}) (4 x^{2} y^{3} y' + y^{4} \frac{d}{d x} [x^{2}])$

Step 2.6

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

$\sec^{2} (x^{2} y^{4}) (4 x^{2} y^{3} y' + y^{4} (2 x))$

Step 2.7

Move $2$ to the left of $y^{4}$ .

$\sec^{2} (x^{2} y^{4}) (4 x^{2} y^{3} y' + 2 \cdot y^{4} x)$

Step 2.8

Simplify.

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Step 2.8.1

Apply the distributive property.

$\sec^{2} (x^{2} y^{4}) (4 x^{2} y^{3} y') + \sec^{2} (x^{2} y^{4}) (2 y^{4} x)$

Step 2.8.2

Reorder terms.

$4 y^{3} x^{2} \sec^{2} (x^{2} y^{4}) y' + 2 y^{4} x \sec^{2} (x^{2} y^{4})$

Step 3

Differentiate the right side of the equation.

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Step 3.1

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

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

Step 3.2

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

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Step 3.2.1

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

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

Step 3.2.2

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

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

Step 3.2.3

Multiply $3$ by $1$ .

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

Step 3.3

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

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Step 3.3.1

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 $g (x) = y$ .

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Step 3.3.1.1

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

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

Step 3.3.1.2

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

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

Step 3.3.1.3

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

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

Step 3.3.2

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

$3 + 2 y y'$

Step 3.4

Reorder terms.

$2 y y' + 3$

Step 4

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

$4 y^{3} x^{2} \sec^{2} (x^{2} y^{4}) y' + 2 y^{4} x \sec^{2} (x^{2} y^{4}) = 2 y y' + 3$

Step 5

Solve for $y'$ .

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Step 5.1

Simplify the left side.

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Step 5.1.1

Reorder factors in $4 y^{3} x^{2} \sec^{2} (x^{2} y^{4}) y' + 2 y^{4} x \sec^{2} (x^{2} y^{4})$ .

$4 y^{3} x^{2} y' \sec^{2} (x^{2} y^{4}) + 2 y^{4} x \sec^{2} (x^{2} y^{4}) = 2 y y' + 3$

Step 5.2

Subtract $2 y y'$ from both sides of the equation.

$4 y^{3} x^{2} y' \sec^{2} (x^{2} y^{4}) + 2 y^{4} x \sec^{2} (x^{2} y^{4}) - 2 y y' = 3$

Step 5.3

Subtract $2 y^{4} x \sec^{2} (x^{2} y^{4})$ from both sides of the equation.

$4 y^{3} x^{2} y' \sec^{2} (x^{2} y^{4}) - 2 y y' = 3 - 2 y^{4} x \sec^{2} (x^{2} y^{4})$

Step 5.4

Factor $2 y y'$ out of $4 y^{3} x^{2} y' \sec^{2} (x^{2} y^{4}) - 2 y y'$ .

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Step 5.4.1

Factor $2 y y'$ out of $4 y^{3} x^{2} y' \sec^{2} (x^{2} y^{4})$ .

$2 y y' (2 y^{2} x^{2} \sec^{2} (x^{2} y^{4})) - 2 y y' = 3 - 2 y^{4} x \sec^{2} (x^{2} y^{4})$

Step 5.4.2

Factor $2 y y'$ out of $- 2 y y'$ .

$2 y y' (2 y^{2} x^{2} \sec^{2} (x^{2} y^{4})) + 2 y y' \cdot - 1 = 3 - 2 y^{4} x \sec^{2} (x^{2} y^{4})$

Step 5.4.3

Factor $2 y y'$ out of $2 y y' (2 y^{2} x^{2} \sec^{2} (x^{2} y^{4})) + 2 y y' \cdot - 1$ .

$2 y y' (2 y^{2} x^{2} \sec^{2} (x^{2} y^{4}) - 1) = 3 - 2 y^{4} x \sec^{2} (x^{2} y^{4})$

Step 5.5

Divide each term in $2 y y' (2 y^{2} x^{2} \sec^{2} (x^{2} y^{4}) - 1) = 3 - 2 y^{4} x \sec^{2} (x^{2} y^{4})$ by $2 y (2 y^{2} x^{2} \sec^{2} (x^{2} y^{4}) - 1)$ and simplify.

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Step 5.5.1

Divide each term in $2 y y' (2 y^{2} x^{2} \sec^{2} (x^{2} y^{4}) - 1) = 3 - 2 y^{4} x \sec^{2} (x^{2} y^{4})$ by $2 y (2 y^{2} x^{2} \sec^{2} (x^{2} y^{4}) - 1)$ .

$\frac{2 y y' (2 y^{2} x^{2} \sec^{2} (x^{2} y^{4}) - 1)}{2 y (2 y^{2} x^{2} \sec^{2} (x^{2} y^{4}) - 1)} = \frac{3}{2 y (2 y^{2} x^{2} \sec^{2} (x^{2} y^{4}) - 1)} + \frac{- 2 y^{4} x \sec^{2} (x^{2} y^{4})}{2 y (2 y^{2} x^{2} \sec^{2} (x^{2} y^{4}) - 1)}$

Step 5.5.2

Simplify the left side.

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Step 5.5.2.1

Cancel the common factor of $2$ .

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Step 5.5.2.1.1

Cancel the common factor.

Step 5.5.2.1.2

Rewrite the expression.

$\frac{y y' (2 y^{2} x^{2} \sec^{2} (x^{2} y^{4}) - 1)}{y (2 y^{2} x^{2} \sec^{2} (x^{2} y^{4}) - 1)} = \frac{3}{2 y (2 y^{2} x^{2} \sec^{2} (x^{2} y^{4}) - 1)} + \frac{- 2 y^{4} x \sec^{2} (x^{2} y^{4})}{2 y (2 y^{2} x^{2} \sec^{2} (x^{2} y^{4}) - 1)}$

Step 5.5.2.2

Cancel the common factor of $y$ .

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Step 5.5.2.2.1

Cancel the common factor.

Step 5.5.2.2.2

Rewrite the expression.

$\frac{y' (2 y^{2} x^{2} \sec^{2} (x^{2} y^{4}) - 1)}{2 y^{2} x^{2} \sec^{2} (x^{2} y^{4}) - 1} = \frac{3}{2 y (2 y^{2} x^{2} \sec^{2} (x^{2} y^{4}) - 1)} + \frac{- 2 y^{4} x \sec^{2} (x^{2} y^{4})}{2 y (2 y^{2} x^{2} \sec^{2} (x^{2} y^{4}) - 1)}$

Step 5.5.2.3

Cancel the common factor of $2 y^{2} x^{2} \sec^{2} (x^{2} y^{4}) - 1$ .

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Step 5.5.2.3.1

Cancel the common factor.

Step 5.5.2.3.2

Divide $y'$ by $1$ .

$y' = \frac{3}{2 y (2 y^{2} x^{2} \sec^{2} (x^{2} y^{4}) - 1)} + \frac{- 2 y^{4} x \sec^{2} (x^{2} y^{4})}{2 y (2 y^{2} x^{2} \sec^{2} (x^{2} y^{4}) - 1)}$

Step 5.5.3

Simplify the right side.

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Step 5.5.3.1

Simplify each term.

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Step 5.5.3.1.1

Cancel the common factor of $- 2$ and $2$ .

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Step 5.5.3.1.1.1

Factor $2$ out of $- 2 y^{4} x \sec^{2} (x^{2} y^{4})$ .

$y' = \frac{3}{2 y (2 y^{2} x^{2} \sec^{2} (x^{2} y^{4}) - 1)} + \frac{2 (- y^{4} x \sec^{2} (x^{2} y^{4}))}{2 y (2 y^{2} x^{2} \sec^{2} (x^{2} y^{4}) - 1)}$

Step 5.5.3.1.1.2

Cancel the common factors.

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Step 5.5.3.1.1.2.1

Factor $2$ out of $2 y (2 y^{2} x^{2} \sec^{2} (x^{2} y^{4}) - 1)$ .

$y' = \frac{3}{2 y (2 y^{2} x^{2} \sec^{2} (x^{2} y^{4}) - 1)} + \frac{2 (- y^{4} x \sec^{2} (x^{2} y^{4}))}{2 (y (2 y^{2} x^{2} \sec^{2} (x^{2} y^{4}) - 1))}$

Step 5.5.3.1.1.2.2

Cancel the common factor.

$y' = \frac{3}{2 y (2 y^{2} x^{2} \sec^{2} (x^{2} y^{4}) - 1)} + \frac{2 (- y^{4} x \sec^{2} (x^{2} y^{4}))}{2 (y (2 y^{2} x^{2} \sec^{2} (x^{2} y^{4}) - 1))}$

Step 5.5.3.1.1.2.3

Rewrite the expression.

$y' = \frac{3}{2 y (2 y^{2} x^{2} \sec^{2} (x^{2} y^{4}) - 1)} + \frac{- y^{4} x \sec^{2} (x^{2} y^{4})}{y (2 y^{2} x^{2} \sec^{2} (x^{2} y^{4}) - 1)}$

Step 5.5.3.1.2

Cancel the common factor of $y^{4}$ and $y$ .

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Step 5.5.3.1.2.1

Factor $y$ out of $- y^{4} x \sec^{2} (x^{2} y^{4})$ .

$y' = \frac{3}{2 y (2 y^{2} x^{2} \sec^{2} (x^{2} y^{4}) - 1)} + \frac{y (- y^{3} x \sec^{2} (x^{2} y^{4}))}{y (2 y^{2} x^{2} \sec^{2} (x^{2} y^{4}) - 1)}$

Step 5.5.3.1.2.2

Cancel the common factors.

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Step 5.5.3.1.2.2.1

Cancel the common factor.

$y' = \frac{3}{2 y (2 y^{2} x^{2} \sec^{2} (x^{2} y^{4}) - 1)} + \frac{y (- y^{3} x \sec^{2} (x^{2} y^{4}))}{y (2 y^{2} x^{2} \sec^{2} (x^{2} y^{4}) - 1)}$

Step 5.5.3.1.2.2.2

Rewrite the expression.

$y' = \frac{3}{2 y (2 y^{2} x^{2} \sec^{2} (x^{2} y^{4}) - 1)} + \frac{- y^{3} x \sec^{2} (x^{2} y^{4})}{2 y^{2} x^{2} \sec^{2} (x^{2} y^{4}) - 1}$

Step 5.5.3.1.3

Move the negative in front of the fraction.

$y' = \frac{3}{2 y (2 y^{2} x^{2} \sec^{2} (x^{2} y^{4}) - 1)} - \frac{y^{3} x \sec^{2} (x^{2} y^{4})}{2 y^{2} x^{2} \sec^{2} (x^{2} y^{4}) - 1}$

Step 5.5.3.2

To write $- \frac{y^{3} x \sec^{2} (x^{2} y^{4})}{2 y^{2} x^{2} \sec^{2} (x^{2} y^{4}) - 1}$ as a fraction with a common denominator, multiply by $\frac{2 y}{2 y}$ .

$y' = \frac{3}{2 y (2 y^{2} x^{2} \sec^{2} (x^{2} y^{4}) - 1)} - \frac{y^{3} x \sec^{2} (x^{2} y^{4})}{2 y^{2} x^{2} \sec^{2} (x^{2} y^{4}) - 1} \cdot \frac{2 y}{2 y}$

Step 5.5.3.3

Write each expression with a common denominator of $2 y (2 y^{2} x^{2} \sec^{2} (x^{2} y^{4}) - 1)$ , by multiplying each by an appropriate factor of $1$ .

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Step 5.5.3.3.1

Multiply $\frac{y^{3} x \sec^{2} (x^{2} y^{4})}{2 y^{2} x^{2} \sec^{2} (x^{2} y^{4}) - 1}$ by $\frac{2 y}{2 y}$ .

$y' = \frac{3}{2 y (2 y^{2} x^{2} \sec^{2} (x^{2} y^{4}) - 1)} - \frac{y^{3} x \sec^{2} (x^{2} y^{4}) (2 y)}{(2 y^{2} x^{2} \sec^{2} (x^{2} y^{4}) - 1) (2 y)}$

Step 5.5.3.3.2

Reorder the factors of $(2 y^{2} x^{2} \sec^{2} (x^{2} y^{4}) - 1) (2 y)$ .

$y' = \frac{3}{2 y (2 y^{2} x^{2} \sec^{2} (x^{2} y^{4}) - 1)} - \frac{y^{3} x \sec^{2} (x^{2} y^{4}) (2 y)}{2 y (2 y^{2} x^{2} \sec^{2} (x^{2} y^{4}) - 1)}$

Step 5.5.3.4

Combine the numerators over the common denominator.

$y' = \frac{3 - y^{3} x \sec^{2} (x^{2} y^{4}) (2 y)}{2 y (2 y^{2} x^{2} \sec^{2} (x^{2} y^{4}) - 1)}$

Step 5.5.3.5

Simplify the numerator.

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Step 5.5.3.5.1

Multiply $y^{3}$ by $y$ by adding the exponents.

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Step 5.5.3.5.1.1

Move $y$ .

$y' = \frac{3 - (y \cdot y^{3}) x \sec^{2} (x^{2} y^{4}) \cdot 2}{2 y (2 y^{2} x^{2} \sec^{2} (x^{2} y^{4}) - 1)}$

Step 5.5.3.5.1.2

Multiply $y$ by $y^{3}$ .

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Step 5.5.3.5.1.2.1

Raise $y$ to the power of $1$ .

$y' = \frac{3 - (y^{1} y^{3}) x \sec^{2} (x^{2} y^{4}) \cdot 2}{2 y (2 y^{2} x^{2} \sec^{2} (x^{2} y^{4}) - 1)}$

Step 5.5.3.5.1.2.2

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

$y' = \frac{3 - y^{1 + 3} x \sec^{2} (x^{2} y^{4}) \cdot 2}{2 y (2 y^{2} x^{2} \sec^{2} (x^{2} y^{4}) - 1)}$

Step 5.5.3.5.1.3

Add $1$ and $3$ .

$y' = \frac{3 - y^{4} x \sec^{2} (x^{2} y^{4}) \cdot 2}{2 y (2 y^{2} x^{2} \sec^{2} (x^{2} y^{4}) - 1)}$

Step 5.5.3.5.2

Multiply $2$ by $- 1$ .

$y' = \frac{3 - 2 y^{4} x \sec^{2} (x^{2} y^{4})}{2 y (2 y^{2} x^{2} \sec^{2} (x^{2} y^{4}) - 1)}$

Step 6

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

$\frac{d y}{d x} = \frac{3 - 2 y^{4} x \sec^{2} (x^{2} y^{4})}{2 y (2 y^{2} x^{2} \sec^{2} (x^{2} y^{4}) - 1)}$