Calculus Examples

$\tan (4 x + y) = 4 x$

Step 1

Differentiate both sides of the equation.

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

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 $g (x) = 4 x + y$ .

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Step 2.1.1

To apply the Chain Rule, set $u$ as $4 x + y$ .

$\frac{d}{d u} [\tan (u)] \frac{d}{d x} [4 x + y]$

Step 2.1.2

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

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

Step 2.1.3

Replace all occurrences of $u$ with $4 x + y$ .

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

Step 2.2

Differentiate.

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Step 2.2.1

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

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

Step 2.2.2

Since $4$ is constant with respect to $x$ , the derivative of $4 x$ with respect to $x$ is $4 \frac{d}{d x} [x]$ .

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

Step 2.2.3

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

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

Step 2.2.4

Multiply $4$ by $1$ .

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

Step 2.3

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

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

Step 3

Differentiate the right side of the equation.

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Step 3.1

Since $4$ is constant with respect to $x$ , the derivative of $4 x$ with respect to $x$ is $4 \frac{d}{d x} [x]$ .

$4 \frac{d}{d x} [x]$

Step 3.2

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

$4 \cdot 1$

Step 3.3

Multiply $4$ by $1$ .

$4$

Step 4

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

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

Step 5

Solve for $y'$ .

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Step 5.1

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

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Step 5.1.1

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

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

Step 5.1.2

Simplify the left side.

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Step 5.1.2.1

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

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Step 5.1.2.1.1

Cancel the common factor.

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

Step 5.1.2.1.2

Divide $4 + y'$ by $1$ .

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

Step 5.2

Subtract $4$ from both sides of the equation.

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

Step 6

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

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