Calculus Examples

$\frac{d}{d u} (9 u^{0.1} - 6 u^{3.4})$

Step 1

This derivative could not be completed using the chain rule. Mathway will use another method.

Step 2

By the Sum Rule, the derivative of $9 u^{0.1} - 6 u^{3.4}$ with respect to $u$ is $\frac{d}{d u} [9 u^{0.1}] + \frac{d}{d u} [- 6 u^{3.4}]$ .

$\frac{d}{d u} [9 u^{0.1}] + \frac{d}{d u} [- 6 u^{3.4}]$

Step 3

Evaluate $\frac{d}{d u} [9 u^{0.1}]$ .

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Step 3.1

Since $9$ is constant with respect to $u$ , the derivative of $9 u^{0.1}$ with respect to $u$ is $9 \frac{d}{d u} [u^{0.1}]$ .

$9 \frac{d}{d u} [u^{0.1}] + \frac{d}{d u} [- 6 u^{3.4}]$

Step 3.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = 0.1$ .

$9 (0.1 u^{- 0.9}) + \frac{d}{d u} [- 6 u^{3.4}]$

Step 3.3

Multiply $0.1$ by $9$ .

$0.9 u^{- 0.9} + \frac{d}{d u} [- 6 u^{3.4}]$

Step 4

Evaluate $\frac{d}{d u} [- 6 u^{3.4}]$ .

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Step 4.1

Since $- 6$ is constant with respect to $u$ , the derivative of $- 6 u^{3.4}$ with respect to $u$ is $- 6 \frac{d}{d u} [u^{3.4}]$ .

$0.9 u^{- 0.9} - 6 \frac{d}{d u} [u^{3.4}]$

Step 4.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = 3.4$ .

$0.9 u^{- 0.9} - 6 (3.4 u^{2.4})$

Step 4.3

Multiply $3.4$ by $- 6$ .

$0.9 u^{- 0.9} - 20.4 u^{2.4}$

Step 5

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

$0.9 \frac{1}{u^{0.9}} - 20.4 u^{2.4}$

Step 6

Simplify.

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Step 6.1

Combine $0.9$ and $\frac{1}{u^{0.9}}$ .

$\frac{0.9}{u^{0.9}} - 20.4 u^{2.4}$

Step 6.2

Reorder terms.

$- 20.4 u^{2.4} + \frac{0.9}{u^{0.9}}$