Calculus Examples

$y = \sin (4 x) - 8 x^{2} + 9 x + π$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (y) = \frac{d}{d x} (\sin (4 x) - 8 x^{2} + 9 x + π)$

Step 2

The derivative of $y$ with respect to $x$ is $y'$ .

$y'$

Step 3

Differentiate the right side of the equation.

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Step 3.1

By the Sum Rule, the derivative of $\sin (4 x) - 8 x^{2} + 9 x + π$ with respect to $x$ is $\frac{d}{d x} [\sin (4 x)] + \frac{d}{d x} [- 8 x^{2}] + \frac{d}{d x} [9 x] + \frac{d}{d x} [π]$ .

$\frac{d}{d x} [\sin (4 x)] + \frac{d}{d x} [- 8 x^{2}] + \frac{d}{d x} [9 x] + \frac{d}{d x} [π]$

Step 3.2

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

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Step 3.2.1

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 $g (x) = 4 x$ .

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Step 3.2.1.1

To apply the Chain Rule, set $u$ as $4 x$ .

$\frac{d}{d u} [\sin (u)] \frac{d}{d x} [4 x] + \frac{d}{d x} [- 8 x^{2}] + \frac{d}{d x} [9 x] + \frac{d}{d x} [π]$

Step 3.2.1.2

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

$\cos (u) \frac{d}{d x} [4 x] + \frac{d}{d x} [- 8 x^{2}] + \frac{d}{d x} [9 x] + \frac{d}{d x} [π]$

Step 3.2.1.3

Replace all occurrences of $u$ with $4 x$ .

$\cos (4 x) \frac{d}{d x} [4 x] + \frac{d}{d x} [- 8 x^{2}] + \frac{d}{d x} [9 x] + \frac{d}{d x} [π]$

Step 3.2.2

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

$\cos (4 x) (4 \frac{d}{d x} [x]) + \frac{d}{d x} [- 8 x^{2}] + \frac{d}{d x} [9 x] + \frac{d}{d x} [π]$

Step 3.2.3

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

$\cos (4 x) (4 \cdot 1) + \frac{d}{d x} [- 8 x^{2}] + \frac{d}{d x} [9 x] + \frac{d}{d x} [π]$

Step 3.2.4

Multiply $4$ by $1$ .

$\cos (4 x) \cdot 4 + \frac{d}{d x} [- 8 x^{2}] + \frac{d}{d x} [9 x] + \frac{d}{d x} [π]$

Step 3.2.5

Move $4$ to the left of $\cos (4 x)$ .

$4 \cos (4 x) + \frac{d}{d x} [- 8 x^{2}] + \frac{d}{d x} [9 x] + \frac{d}{d x} [π]$

Step 3.3

Evaluate $\frac{d}{d x} [- 8 x^{2}]$ .

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Step 3.3.1

Since $- 8$ is constant with respect to $x$ , the derivative of $- 8 x^{2}$ with respect to $x$ is $- 8 \frac{d}{d x} [x^{2}]$ .

$4 \cos (4 x) - 8 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [9 x] + \frac{d}{d x} [π]$

Step 3.3.2

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

$4 \cos (4 x) - 8 (2 x) + \frac{d}{d x} [9 x] + \frac{d}{d x} [π]$

Step 3.3.3

Multiply $2$ by $- 8$ .

$4 \cos (4 x) - 16 x + \frac{d}{d x} [9 x] + \frac{d}{d x} [π]$

Step 3.4

Evaluate $\frac{d}{d x} [9 x]$ .

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Step 3.4.1

Since $9$ is constant with respect to $x$ , the derivative of $9 x$ with respect to $x$ is $9 \frac{d}{d x} [x]$ .

$4 \cos (4 x) - 16 x + 9 \frac{d}{d x} [x] + \frac{d}{d x} [π]$

Step 3.4.2

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

$4 \cos (4 x) - 16 x + 9 \cdot 1 + \frac{d}{d x} [π]$

Step 3.4.3

Multiply $9$ by $1$ .

$4 \cos (4 x) - 16 x + 9 + \frac{d}{d x} [π]$

Step 3.5

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

$4 \cos (4 x) - 16 x + 9 + 0$

Step 3.6

Simplify.

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Step 3.6.1

Add $4 \cos (4 x) - 16 x + 9$ and $0$ .

$4 \cos (4 x) - 16 x + 9$

Step 3.6.2

Reorder terms.

$- 16 x + 9 + 4 \cos (4 x)$

Step 4

Reform the equation by setting the left side equal to the right side.

$y' = - 16 x + 9 + 4 \cos (4 x)$

Step 5

Replace $y'$ with $\frac{d y}{d x}$ .

$\frac{d y}{d x} = - 16 x + 9 + 4 \cos (4 x)$