Calculus Examples

$\sin (x + y) = 4 x - 4 y$

Step 1

Set $\sin (x + y)$ as a function of $x$ .

$f (x) = 4 x - 4 y$

Step 2

Find the derivative.

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Step 2.1

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

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

Step 2.2

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

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Step 2.2.1

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

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

Step 2.2.2

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

$4 \cdot 1 + \frac{d}{d x} [- 4 y]$

Step 2.2.3

Multiply $4$ by $1$ .

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

Step 2.3

Differentiate using the Constant Rule.

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Step 2.3.1

Since $- 4 y$ is constant with respect to $x$ , the derivative of $- 4 y$ with respect to $x$ is $0$ .

$4 + 0$

Step 2.3.2

Add $4$ and $0$ .

$4$

Step 3

Since $4 \neq 0$ , there are no solutions.

No solution

Step 4

There are no solution found by setting the derivative equal to $0$ , $4 = 0$ so there are no horizontal tangent lines.

No horizontal tangent lines found

Step 5