Calculus Examples

$f (x) = \sin (x) - \tan (x) + x^{2} + \frac{1}{x^{2}} - \frac{1}{\sqrt{x}}$

Step 1

By the Sum Rule, the derivative of $\sin (x) - \tan (x) + x^{2} + \frac{1}{x^{2}} - \frac{1}{\sqrt{x}}$ with respect to $x$ is $\frac{d}{d x} [\sin (x)] + \frac{d}{d x} [- \tan (x)] + \frac{d}{d x} [x^{2}] + \frac{d}{d x} [\frac{1}{x^{2}}] + \frac{d}{d x} [- \frac{1}{\sqrt{x}}]$ .

$\frac{d}{d x} [\sin (x)] + \frac{d}{d x} [- \tan (x)] + \frac{d}{d x} [x^{2}] + \frac{d}{d x} [\frac{1}{x^{2}}] + \frac{d}{d x} [- \frac{1}{\sqrt{x}}]$

Step 2

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

$\cos (x) + \frac{d}{d x} [- \tan (x)] + \frac{d}{d x} [x^{2}] + \frac{d}{d x} [\frac{1}{x^{2}}] + \frac{d}{d x} [- \frac{1}{\sqrt{x}}]$

Step 3

Evaluate $\frac{d}{d x} [- \tan (x)]$ .

Tap for more steps...

Step 3.1

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

$\cos (x) - \frac{d}{d x} [\tan (x)] + \frac{d}{d x} [x^{2}] + \frac{d}{d x} [\frac{1}{x^{2}}] + \frac{d}{d x} [- \frac{1}{\sqrt{x}}]$

Step 3.2

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

$\cos (x) - \sec^{2} (x) + \frac{d}{d x} [x^{2}] + \frac{d}{d x} [\frac{1}{x^{2}}] + \frac{d}{d x} [- \frac{1}{\sqrt{x}}]$

Step 4

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

$\cos (x) - \sec^{2} (x) + 2 x + \frac{d}{d x} [\frac{1}{x^{2}}] + \frac{d}{d x} [- \frac{1}{\sqrt{x}}]$

Step 5

Evaluate $\frac{d}{d x} [\frac{1}{x^{2}}]$ .

Tap for more steps...

Step 5.1

Rewrite $\frac{1}{x^{2}}$ as ${(x^{2})}^{- 1}$ .

$\cos (x) - \sec^{2} (x) + 2 x + \frac{d}{d x} [{(x^{2})}^{- 1}] + \frac{d}{d x} [- \frac{1}{\sqrt{x}}]$

Step 5.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{- 1}$ and $g (x) = x^{2}$ .

Tap for more steps...

Step 5.2.1

To apply the Chain Rule, set $u_{1}$ as $x^{2}$ .

$\cos (x) - \sec^{2} (x) + 2 x + \frac{d}{d u_{1}} [{u_{1}}^{- 1}] \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- \frac{1}{\sqrt{x}}]$

Step 5.2.2

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

$\cos (x) - \sec^{2} (x) + 2 x - {u_{1}}^{- 2} \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- \frac{1}{\sqrt{x}}]$

Step 5.2.3

Replace all occurrences of $u_{1}$ with $x^{2}$ .

$\cos (x) - \sec^{2} (x) + 2 x - {(x^{2})}^{- 2} \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- \frac{1}{\sqrt{x}}]$

Step 5.3

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

$\cos (x) - \sec^{2} (x) + 2 x - {(x^{2})}^{- 2} (2 x) + \frac{d}{d x} [- \frac{1}{\sqrt{x}}]$

Step 5.4

Multiply the exponents in ${(x^{2})}^{- 2}$ .

Tap for more steps...

Step 5.4.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Step 5.4.2

Multiply $2$ by $- 2$ .

$\cos (x) - \sec^{2} (x) + 2 x - x^{- 4} (2 x) + \frac{d}{d x} [- \frac{1}{\sqrt{x}}]$

Step 5.5

Multiply $2$ by $- 1$ .

$\cos (x) - \sec^{2} (x) + 2 x - 2 x^{- 4} x + \frac{d}{d x} [- \frac{1}{\sqrt{x}}]$

Step 5.6

Raise $x$ to the power of $1$ .

$\cos (x) - \sec^{2} (x) + 2 x - 2 (x^{1} x^{- 4}) + \frac{d}{d x} [- \frac{1}{\sqrt{x}}]$

Step 5.7

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\cos (x) - \sec^{2} (x) + 2 x - 2 x^{1 - 4} + \frac{d}{d x} [- \frac{1}{\sqrt{x}}]$

Step 5.8

Subtract $4$ from $1$ .

$\cos (x) - \sec^{2} (x) + 2 x - 2 x^{- 3} + \frac{d}{d x} [- \frac{1}{\sqrt{x}}]$

Step 6

Evaluate $\frac{d}{d x} [- \frac{1}{\sqrt{x}}]$ .

Tap for more steps...

Step 6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x}$ as $x^{\frac{1}{2}}$ .

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

Step 6.2

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = - 1$ and $g (x) = \frac{1}{x^{\frac{1}{2}}}$ .

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

Step 6.3

Rewrite $\frac{1}{x^{\frac{1}{2}}}$ as ${(x^{\frac{1}{2}})}^{- 1}$ .

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

Step 6.4

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{- 1}$ and $g (x) = x^{\frac{1}{2}}$ .

Tap for more steps...

Step 6.4.1

To apply the Chain Rule, set $u_{2}$ as $x^{\frac{1}{2}}$ .

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

Step 6.4.2

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

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

Step 6.4.3

Replace all occurrences of $u_{2}$ with $x^{\frac{1}{2}}$ .

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

Step 6.5

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

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

Step 6.6

Since $- 1$ is constant with respect to $x$ , the derivative of $- 1$ with respect to $x$ is $0$ .

$\cos (x) - \sec^{2} (x) + 2 x - 2 x^{- 3} - (- {(x^{\frac{1}{2}})}^{- 2} (\frac{1}{2} x^{\frac{1}{2} - 1})) + \frac{1}{x^{\frac{1}{2}}} \cdot 0$

Step 6.7

Multiply the exponents in ${(x^{\frac{1}{2}})}^{- 2}$ .

Tap for more steps...

Step 6.7.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\cos (x) - \sec^{2} (x) + 2 x - 2 x^{- 3} - (- x^{\frac{1}{2} \cdot - 2} (\frac{1}{2} x^{\frac{1}{2} - 1})) + \frac{1}{x^{\frac{1}{2}}} \cdot 0$

Step 6.7.2

Cancel the common factor of $2$ .

Tap for more steps...

Step 6.7.2.1

Factor $2$ out of $- 2$ .

$\cos (x) - \sec^{2} (x) + 2 x - 2 x^{- 3} - (- x^{\frac{1}{2} \cdot (2 (- 1))} (\frac{1}{2} x^{\frac{1}{2} - 1})) + \frac{1}{x^{\frac{1}{2}}} \cdot 0$

Step 6.7.2.2

Cancel the common factor.

$\cos (x) - \sec^{2} (x) + 2 x - 2 x^{- 3} - (- x^{\frac{1}{2} \cdot (2 \cdot - 1)} (\frac{1}{2} x^{\frac{1}{2} - 1})) + \frac{1}{x^{\frac{1}{2}}} \cdot 0$

Step 6.7.2.3

Rewrite the expression.

$\cos (x) - \sec^{2} (x) + 2 x - 2 x^{- 3} - (- x^{- 1} (\frac{1}{2} x^{\frac{1}{2} - 1})) + \frac{1}{x^{\frac{1}{2}}} \cdot 0$

Step 6.8

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\cos (x) - \sec^{2} (x) + 2 x - 2 x^{- 3} - (- x^{- 1} (\frac{1}{2} x^{\frac{1}{2} - 1 \cdot \frac{2}{2}})) + \frac{1}{x^{\frac{1}{2}}} \cdot 0$

Step 6.9

Combine $- 1$ and $\frac{2}{2}$ .

$\cos (x) - \sec^{2} (x) + 2 x - 2 x^{- 3} - (- x^{- 1} (\frac{1}{2} x^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}})) + \frac{1}{x^{\frac{1}{2}}} \cdot 0$

Step 6.10

Combine the numerators over the common denominator.

$\cos (x) - \sec^{2} (x) + 2 x - 2 x^{- 3} - (- x^{- 1} (\frac{1}{2} x^{\frac{1 - 1 \cdot 2}{2}})) + \frac{1}{x^{\frac{1}{2}}} \cdot 0$

Step 6.11

Simplify the numerator.

Tap for more steps...

Step 6.11.1

Multiply $- 1$ by $2$ .

$\cos (x) - \sec^{2} (x) + 2 x - 2 x^{- 3} - (- x^{- 1} (\frac{1}{2} x^{\frac{1 - 2}{2}})) + \frac{1}{x^{\frac{1}{2}}} \cdot 0$

Step 6.11.2

Subtract $2$ from $1$ .

$\cos (x) - \sec^{2} (x) + 2 x - 2 x^{- 3} - (- x^{- 1} (\frac{1}{2} x^{\frac{- 1}{2}})) + \frac{1}{x^{\frac{1}{2}}} \cdot 0$

Step 6.12

Move the negative in front of the fraction.

$\cos (x) - \sec^{2} (x) + 2 x - 2 x^{- 3} - (- x^{- 1} (\frac{1}{2} x^{- \frac{1}{2}})) + \frac{1}{x^{\frac{1}{2}}} \cdot 0$

Step 6.13

Combine $\frac{1}{2}$ and $x^{- \frac{1}{2}}$ .

$\cos (x) - \sec^{2} (x) + 2 x - 2 x^{- 3} - (- x^{- 1} \frac{x^{- \frac{1}{2}}}{2}) + \frac{1}{x^{\frac{1}{2}}} \cdot 0$

Step 6.14

Combine $\frac{x^{- \frac{1}{2}}}{2}$ and $x^{- 1}$ .

$\cos (x) - \sec^{2} (x) + 2 x - 2 x^{- 3} - - \frac{x^{- \frac{1}{2}} x^{- 1}}{2} + \frac{1}{x^{\frac{1}{2}}} \cdot 0$

Step 6.15

Multiply $x^{- \frac{1}{2}}$ by $x^{- 1}$ by adding the exponents.

Tap for more steps...

Step 6.15.1

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\cos (x) - \sec^{2} (x) + 2 x - 2 x^{- 3} - - \frac{x^{- \frac{1}{2} - 1}}{2} + \frac{1}{x^{\frac{1}{2}}} \cdot 0$

Step 6.15.2

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\cos (x) - \sec^{2} (x) + 2 x - 2 x^{- 3} - - \frac{x^{- \frac{1}{2} - 1 \cdot \frac{2}{2}}}{2} + \frac{1}{x^{\frac{1}{2}}} \cdot 0$

Step 6.15.3

Combine $- 1$ and $\frac{2}{2}$ .

$\cos (x) - \sec^{2} (x) + 2 x - 2 x^{- 3} - - \frac{x^{- \frac{1}{2} + \frac{- 1 \cdot 2}{2}}}{2} + \frac{1}{x^{\frac{1}{2}}} \cdot 0$

Step 6.15.4

Combine the numerators over the common denominator.

$\cos (x) - \sec^{2} (x) + 2 x - 2 x^{- 3} - - \frac{x^{\frac{- 1 - 1 \cdot 2}{2}}}{2} + \frac{1}{x^{\frac{1}{2}}} \cdot 0$

Step 6.15.5

Simplify the numerator.

Tap for more steps...

Step 6.15.5.1

Multiply $- 1$ by $2$ .

$\cos (x) - \sec^{2} (x) + 2 x - 2 x^{- 3} - - \frac{x^{\frac{- 1 - 2}{2}}}{2} + \frac{1}{x^{\frac{1}{2}}} \cdot 0$

Step 6.15.5.2

Subtract $2$ from $- 1$ .

$\cos (x) - \sec^{2} (x) + 2 x - 2 x^{- 3} - - \frac{x^{\frac{- 3}{2}}}{2} + \frac{1}{x^{\frac{1}{2}}} \cdot 0$

Step 6.15.6

Move the negative in front of the fraction.

$\cos (x) - \sec^{2} (x) + 2 x - 2 x^{- 3} - - \frac{x^{- \frac{3}{2}}}{2} + \frac{1}{x^{\frac{1}{2}}} \cdot 0$

Step 6.16

Move $x^{- \frac{3}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\cos (x) - \sec^{2} (x) + 2 x - 2 x^{- 3} - - \frac{1}{2 x^{\frac{3}{2}}} + \frac{1}{x^{\frac{1}{2}}} \cdot 0$

Step 6.17

Multiply $- 1$ by $- 1$ .

$\cos (x) - \sec^{2} (x) + 2 x - 2 x^{- 3} + 1 \frac{1}{2 x^{\frac{3}{2}}} + \frac{1}{x^{\frac{1}{2}}} \cdot 0$

Step 6.18

Multiply $\frac{1}{2 x^{\frac{3}{2}}}$ by $1$ .

$\cos (x) - \sec^{2} (x) + 2 x - 2 x^{- 3} + \frac{1}{2 x^{\frac{3}{2}}} + \frac{1}{x^{\frac{1}{2}}} \cdot 0$

Step 6.19

Multiply $\frac{1}{x^{\frac{1}{2}}}$ by $0$ .

$\cos (x) - \sec^{2} (x) + 2 x - 2 x^{- 3} + \frac{1}{2 x^{\frac{3}{2}}} + 0$

Step 6.20

Add $\frac{1}{2 x^{\frac{3}{2}}}$ and $0$ .

$\cos (x) - \sec^{2} (x) + 2 x - 2 x^{- 3} + \frac{1}{2 x^{\frac{3}{2}}}$

Step 7

Simplify.

Tap for more steps...

Step 7.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\cos (x) - \sec^{2} (x) + 2 x - 2 \frac{1}{x^{3}} + \frac{1}{2 x^{\frac{3}{2}}}$

Step 7.2

Combine terms.

Tap for more steps...

Step 7.2.1

Combine $- 2$ and $\frac{1}{x^{3}}$ .

$\cos (x) - \sec^{2} (x) + 2 x + \frac{- 2}{x^{3}} + \frac{1}{2 x^{\frac{3}{2}}}$

Step 7.2.2

Move the negative in front of the fraction.

$\cos (x) - \sec^{2} (x) + 2 x - \frac{2}{x^{3}} + \frac{1}{2 x^{\frac{3}{2}}}$

Step 7.3

Reorder terms.

$2 x - \frac{2}{x^{3}} + \frac{1}{2 x^{\frac{3}{2}}} + \cos (x) - \sec^{2} (x)$

Step 7.4

Simplify each term.

Tap for more steps...

Step 7.4.1

Rewrite $\sec (x)$ in terms of sines and cosines.

$2 x - \frac{2}{x^{3}} + \frac{1}{2 x^{\frac{3}{2}}} + \cos (x) - {(\frac{1}{\cos (x)})}^{2}$

Step 7.4.2

Apply the product rule to $\frac{1}{\cos (x)}$ .

$2 x - \frac{2}{x^{3}} + \frac{1}{2 x^{\frac{3}{2}}} + \cos (x) - \frac{1^{2}}{\cos^{2} (x)}$

Step 7.4.3

One to any power is one.

$2 x - \frac{2}{x^{3}} + \frac{1}{2 x^{\frac{3}{2}}} + \cos (x) - \frac{1}{\cos^{2} (x)}$

Step 7.5

Simplify each term.

Tap for more steps...

Step 7.5.1

Rewrite $1$ as $1^{2}$ .

$2 x - \frac{2}{x^{3}} + \frac{1}{2 x^{\frac{3}{2}}} + \cos (x) - \frac{1^{2}}{\cos^{2} (x)}$

Step 7.5.2

Rewrite $\frac{1^{2}}{\cos^{2} (x)}$ as ${(\frac{1}{\cos (x)})}^{2}$ .

$2 x - \frac{2}{x^{3}} + \frac{1}{2 x^{\frac{3}{2}}} + \cos (x) - {(\frac{1}{\cos (x)})}^{2}$

Step 7.5.3

Convert from $\frac{1}{\cos (x)}$ to $\sec (x)$ .

$2 x - \frac{2}{x^{3}} + \frac{1}{2 x^{\frac{3}{2}}} + \cos (x) - \sec^{2} (x)$