Algebra Examples

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

Step 1

Differentiate both sides of the equation.

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

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 Quotient Rule which states that $\frac{d}{d x} [\frac{f (x)}{g (x)}]$ is $\frac{g (x) \frac{d}{d x} [f (x)] - f (x) \frac{d}{d x} [g (x)]}{{g (x)}^{2}}$ where $f (x) = x^{2}$ and $g (x) = \cos (x)$ .

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

Step 3.2

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

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

Step 3.3

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

$\frac{\cos (x) (2 x) - x^{2} (- \sin (x))}{\cos^{2} (x)}$

Step 3.4

Multiply.

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Step 3.4.1

Multiply $- 1$ by $- 1$ .

$\frac{\cos (x) (2 x) + 1 x^{2} \sin (x)}{\cos^{2} (x)}$

Step 3.4.2

Multiply $x^{2}$ by $1$ .

$\frac{\cos (x) (2 x) + x^{2} \sin (x)}{\cos^{2} (x)}$

Step 3.5

Simplify.

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Step 3.5.1

Simplify the numerator.

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Step 3.5.1.1

Rewrite using the commutative property of multiplication.

$\frac{2 \cos (x) x + x^{2} \sin (x)}{\cos^{2} (x)}$

Step 3.5.1.2

Reorder factors in $2 \cos (x) x + x^{2} \sin (x)$ .

$\frac{2 x \cos (x) + x^{2} \sin (x)}{\cos^{2} (x)}$

Step 3.5.2

Factor $x$ out of $2 x \cos (x) + x^{2} \sin (x)$ .

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Step 3.5.2.1

Factor $x$ out of $2 x \cos (x)$ .

$\frac{x (2 \cos (x)) + x^{2} \sin (x)}{\cos^{2} (x)}$

Step 3.5.2.2

Factor $x$ out of $x^{2} \sin (x)$ .

$\frac{x (2 \cos (x)) + x (x \sin (x))}{\cos^{2} (x)}$

Step 3.5.2.3

Factor $x$ out of $x (2 \cos (x)) + x (x \sin (x))$ .

$\frac{x (2 \cos (x) + x \sin (x))}{\cos^{2} (x)}$

Step 4

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

$y' = \frac{x (2 \cos (x) + x \sin (x))}{\cos^{2} (x)}$

Step 5

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

$\frac{d y}{d x} = \frac{x (2 \cos (x) + x \sin (x))}{\cos^{2} (x)}$