Calculus Examples

$y = \tan (x) + 7 x^{2} - π x + 2$

Step 1

By the Sum Rule, the derivative of $\tan (x) + 7 x^{2} - π x + 2$ with respect to $x$ is $\frac{d}{d x} [\tan (x)] + \frac{d}{d x} [7 x^{2}] + \frac{d}{d x} [- π x] + \frac{d}{d x} [2]$ .

$\frac{d}{d x} [\tan (x)] + \frac{d}{d x} [7 x^{2}] + \frac{d}{d x} [- π x] + \frac{d}{d x} [2]$

Step 2

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

$\sec^{2} (x) + \frac{d}{d x} [7 x^{2}] + \frac{d}{d x} [- π x] + \frac{d}{d x} [2]$

Step 3

Evaluate $\frac{d}{d x} [7 x^{2}]$ .

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Step 3.1

Since $7$ is constant with respect to $x$ , the derivative of $7 x^{2}$ with respect to $x$ is $7 \frac{d}{d x} [x^{2}]$ .

$\sec^{2} (x) + 7 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- π x] + \frac{d}{d x} [2]$

Step 3.2

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

$\sec^{2} (x) + 7 (2 x) + \frac{d}{d x} [- π x] + \frac{d}{d x} [2]$

Step 3.3

Multiply $2$ by $7$ .

$\sec^{2} (x) + 14 x + \frac{d}{d x} [- π x] + \frac{d}{d x} [2]$

Step 4

Evaluate $\frac{d}{d x} [- π x]$ .

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Step 4.1

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

$\sec^{2} (x) + 14 x - π \frac{d}{d x} [x] + \frac{d}{d x} [2]$

Step 4.2

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

$\sec^{2} (x) + 14 x - π \cdot 1 + \frac{d}{d x} [2]$

Step 4.3

Multiply $- 1$ by $1$ .

$\sec^{2} (x) + 14 x - π + \frac{d}{d x} [2]$

Step 5

Since $2$ is constant with respect to $x$ , the derivative of $2$ with respect to $x$ is $0$ .

$\sec^{2} (x) + 14 x - π + 0$

Step 6

Simplify.

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Step 6.1

Add $\sec^{2} (x) + 14 x - π$ and $0$ .

$\sec^{2} (x) + 14 x - π$

Step 6.2

Reorder terms.

$14 x - π + \sec^{2} (x)$