Calculus Examples

$f (x) = 4 x + \cos (π) x$

Step 1

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because cosine is negative in the second quadrant.

$\frac{d}{d x} [4 x - \cos (0) x]$

Step 2

The exact value of $\cos (0)$ is $1$ .

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

Step 3

Differentiate using the Sum Rule.

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Step 3.1

Multiply $- 1$ by $1$ .

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

Step 3.2

By the Sum Rule, the derivative of $4 x - 1 x$ with respect to $x$ is $\frac{d}{d x} [4 x] + \frac{d}{d x} [- 1 x]$ .

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

Step 4

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

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Step 4.1

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

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

Step 4.2

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

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

Step 4.3

Multiply $4$ by $1$ .

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

Step 5

Evaluate $\frac{d}{d x} [- 1 x]$ .

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Step 5.1

Rewrite $- 1 x$ as $- x$ .

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

Step 5.2

Since $- 1$ is constant with respect to $x$ , the derivative of $- x$ with respect to $x$ is $- \frac{d}{d x} [x]$ .

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

Step 5.3

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

$4 - 1 \cdot 1$

Step 5.4

Multiply $- 1$ by $1$ .

$4 - 1$

Step 6

Subtract $1$ from $4$ .

$3$