Calculus Examples

$y = \cos (5 x^{2} - x)$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (y) = \frac{d}{d x} (\cos (5 x^{2} - x))$

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

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 $g (x) = 5 x^{2} - x$ .

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Step 3.1.1

To apply the Chain Rule, set $u$ as $5 x^{2} - x$ .

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

Step 3.1.2

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

$- \sin (u) \frac{d}{d x} [5 x^{2} - x]$

Step 3.1.3

Replace all occurrences of $u$ with $5 x^{2} - x$ .

$- \sin (5 x^{2} - x) \frac{d}{d x} [5 x^{2} - x]$

Step 3.2

Differentiate.

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Step 3.2.1

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

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

Step 3.2.2

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

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

Step 3.2.3

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

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

Step 3.2.4

Multiply $2$ by $5$ .

$- \sin (5 x^{2} - x) (10 x + \frac{d}{d x} [- x])$

Step 3.2.5

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

$- \sin (5 x^{2} - x) (10 x - \frac{d}{d x} [x])$

Step 3.2.6

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

$- \sin (5 x^{2} - x) (10 x - 1 \cdot 1)$

Step 3.2.7

Multiply $- 1$ by $1$ .

$- \sin (5 x^{2} - x) (10 x - 1)$

Step 4

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

$y' = - \sin (5 x^{2} - x) (10 x - 1)$

Step 5

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

$\frac{d y}{d x} = - \sin (5 x^{2} - x) (10 x - 1)$