Calculus Examples

$3 x - 5 \cos^{2} (π x)$

Step 1

By the Sum Rule, the derivative of $3 x - 5 \cos^{2} (π x)$ with respect to $x$ is $\frac{d}{d x} [3 x] + \frac{d}{d x} [- 5 \cos^{2} (π x)]$ .

$\frac{d}{d x} [3 x] + \frac{d}{d x} [- 5 \cos^{2} (π x)]$

Step 2

Evaluate $\frac{d}{d x} [3 x]$ .

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Step 2.1

Since $3$ is constant with respect to $x$ , the derivative of $3 x$ with respect to $x$ is $3 \frac{d}{d x} [x]$ .

$3 \frac{d}{d x} [x] + \frac{d}{d x} [- 5 \cos^{2} (π x)]$

Step 2.2

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

$3 \cdot 1 + \frac{d}{d x} [- 5 \cos^{2} (π x)]$

Step 2.3

Multiply $3$ by $1$ .

$3 + \frac{d}{d x} [- 5 \cos^{2} (π x)]$

Step 3

Evaluate $\frac{d}{d x} [- 5 \cos^{2} (π x)]$ .

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Step 3.1

Since $- 5$ is constant with respect to $x$ , the derivative of $- 5 \cos^{2} (π x)$ with respect to $x$ is $- 5 \frac{d}{d x} [\cos^{2} (π x)]$ .

$3 - 5 \frac{d}{d x} [\cos^{2} (π x)]$

Step 3.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{2}$ and $g (x) = \cos (π x)$ .

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Step 3.2.1

To apply the Chain Rule, set $u_{1}$ as $\cos (π x)$ .

$3 - 5 (\frac{d}{d u_{1}} [{u_{1}}^{2}] \frac{d}{d x} [\cos (π x)])$

Step 3.2.2

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

$3 - 5 (2 u_{1} \frac{d}{d x} [\cos (π x)])$

Step 3.2.3

Replace all occurrences of $u_{1}$ with $\cos (π x)$ .

$3 - 5 (2 \cos (π x) \frac{d}{d x} [\cos (π x)])$

Step 3.3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \cos (x)$ and $g (x) = π x$ .

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Step 3.3.1

To apply the Chain Rule, set $u_{2}$ as $π x$ .

$3 - 5 (2 \cos (π x) (\frac{d}{d u_{2}} [\cos (u_{2})] \frac{d}{d x} [π x]))$

Step 3.3.2

The derivative of $\cos (u_{2})$ with respect to $u_{2}$ is $- \sin (u_{2})$ .

$3 - 5 (2 \cos (π x) (- \sin (u_{2}) \frac{d}{d x} [π x]))$

Step 3.3.3

Replace all occurrences of $u_{2}$ with $π x$ .

$3 - 5 (2 \cos (π x) (- \sin (π x) \frac{d}{d x} [π x]))$

Step 3.4

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

$3 - 5 (2 \cos (π x) (- \sin (π x) (π \frac{d}{d x} [x])))$

Step 3.5

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

$3 - 5 (2 \cos (π x) (- \sin (π x) (π \cdot 1)))$

Step 3.6

Multiply $π$ by $1$ .

$3 - 5 (2 \cos (π x) (- \sin (π x) π))$

Step 3.7

Multiply $- 1$ by $2$ .

$3 - 5 (- 2 \cos (π x) (\sin (π x) π))$

Step 3.8

Multiply $- 2$ by $- 5$ .

$3 + 10 \cos (π x) \sin (π x) π$

Step 4

Reorder terms.

$10 π \cos (π x) \sin (π x) + 3$