Calculus Examples

$\sin (2 x) - \tan (x) - x$

Step 1

By the Sum Rule, the derivative of $\sin (2 x) - \tan (x) - x$ with respect to $x$ is $\frac{d}{d x} [\sin (2 x)] + \frac{d}{d x} [- \tan (x)] + \frac{d}{d x} [- x]$ .

$\frac{d}{d x} [\sin (2 x)] + \frac{d}{d x} [- \tan (x)] + \frac{d}{d x} [- x]$

Step 2

Evaluate $\frac{d}{d x} [\sin (2 x)]$ .

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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 $f (x) = \sin (x)$ and $g (x) = 2 x$ .

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Step 2.1.1

To apply the Chain Rule, set $u$ as $2 x$ .

$\frac{d}{d u} [\sin (u)] \frac{d}{d x} [2 x] + \frac{d}{d x} [- \tan (x)] + \frac{d}{d x} [- x]$

Step 2.1.2

The derivative of $\sin (u)$ with respect to $u$ is $\cos (u)$ .

$\cos (u) \frac{d}{d x} [2 x] + \frac{d}{d x} [- \tan (x)] + \frac{d}{d x} [- x]$

Step 2.1.3

Replace all occurrences of $u$ with $2 x$ .

$\cos (2 x) \frac{d}{d x} [2 x] + \frac{d}{d x} [- \tan (x)] + \frac{d}{d x} [- x]$

Step 2.2

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

$\cos (2 x) (2 \frac{d}{d x} [x]) + \frac{d}{d x} [- \tan (x)] + \frac{d}{d x} [- x]$

Step 2.3

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

$\cos (2 x) (2 \cdot 1) + \frac{d}{d x} [- \tan (x)] + \frac{d}{d x} [- x]$

Step 2.4

Multiply $2$ by $1$ .

$\cos (2 x) \cdot 2 + \frac{d}{d x} [- \tan (x)] + \frac{d}{d x} [- x]$

Step 2.5

Move $2$ to the left of $\cos (2 x)$ .

$2 \cos (2 x) + \frac{d}{d x} [- \tan (x)] + \frac{d}{d x} [- x]$

Step 3

Evaluate $\frac{d}{d x} [- \tan (x)]$ .

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Step 3.1

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

$2 \cos (2 x) - \frac{d}{d x} [\tan (x)] + \frac{d}{d x} [- x]$

Step 3.2

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

$2 \cos (2 x) - \sec^{2} (x) + \frac{d}{d x} [- x]$

Step 4

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

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Step 4.1

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

$2 \cos (2 x) - \sec^{2} (x) - \frac{d}{d x} [x]$

Step 4.2

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

$2 \cos (2 x) - \sec^{2} (x) - 1 \cdot 1$

Step 4.3

Multiply $- 1$ by $1$ .

$2 \cos (2 x) - \sec^{2} (x) - 1$

Step 5

Reorder terms.

$- 1 + 2 \cos (2 x) - \sec^{2} (x)$