Calculus Examples

$355 + \frac{{0.4}^{\frac{1}{2}}}{x}$

Step 1

Differentiate.

Tap for more steps...

Step 1.1

By the Sum Rule, the derivative of $355 + \frac{{0.4}^{\frac{1}{2}}}{x}$ with respect to $x$ is $\frac{d}{d x} [355] + \frac{d}{d x} [\frac{{0.4}^{\frac{1}{2}}}{x}]$ .

$\frac{d}{d x} [355] + \frac{d}{d x} [\frac{{0.4}^{\frac{1}{2}}}{x}]$

Step 1.2

Since $355$ is constant with respect to $x$ , the derivative of $355$ with respect to $x$ is $0$ .

$0 + \frac{d}{d x} [\frac{{0.4}^{\frac{1}{2}}}{x}]$

Step 2

Evaluate $\frac{d}{d x} [\frac{{0.4}^{\frac{1}{2}}}{x}]$ .

Tap for more steps...

Step 2.1

Rewrite $0.4$ as ${0.63245553}^{2}$ .

$0 + \frac{d}{d x} [\frac{{({0.63245553}^{2})}^{\frac{1}{2}}}{x}]$

Step 2.2

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

$0 + \frac{d}{d x} [\frac{{0.63245553}^{2 (\frac{1}{2})}}{x}]$

Step 2.3

Cancel the common factor of $2$ .

Tap for more steps...

Step 2.3.1

Cancel the common factor.

$0 + \frac{d}{d x} [\frac{{0.63245553}^{2 (\frac{1}{2})}}{x}]$

Step 2.3.2

Rewrite the expression.

$0 + \frac{d}{d x} [\frac{{0.63245553}^{1}}{x}]$

Step 2.4

Evaluate the exponent.

$0 + \frac{d}{d x} [\frac{0.63245553}{x}]$

Step 2.5

Since $0.63245553$ is constant with respect to $x$ , the derivative of $\frac{0.63245553}{x}$ with respect to $x$ is $0.63245553 \frac{d}{d x} [\frac{1}{x}]$ .

$0 + 0.63245553 \frac{d}{d x} [\frac{1}{x}]$

Step 2.6

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

$0 + 0.63245553 \frac{d}{d x} [x^{- 1}]$

Step 2.7

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

$0 + 0.63245553 (- x^{- 2})$

Step 2.8

Multiply $- 1$ by $0.63245553$ .

$0 - 0.63245553 x^{- 2}$

Step 3

Simplify.

Tap for more steps...

Step 3.1

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

$0 - 0.63245553 \frac{1}{x^{2}}$

Step 3.2

Combine terms.

Tap for more steps...

Step 3.2.1

Combine $- 0.63245553$ and $\frac{1}{x^{2}}$ .

$0 + \frac{- 0.63245553}{x^{2}}$

Step 3.2.2

Move the negative in front of the fraction.

$0 - \frac{0.63245553}{x^{2}}$

Step 3.2.3

Subtract $\frac{0.63245553}{x^{2}}$ from $0$ .

$- \frac{0.63245553}{x^{2}}$