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Find the derivative of $f(x)$ . $\newline$ $f(x) = \frac{1}{x^3}$ $\newline$ Write your answer as a constant times a power of $x$ . $\newline$ $f'(x) =$ ______

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Q. Find the derivative of

f(x)

.

\newline

f(x) = \frac{1}{x^3}

\newline

Write your answer as a constant times a power of

x

.

\newline

f'(x) =

______

Rewrite function as single power: Rewrite the function as a single power of $x$ . The function $f(x) = \frac{1}{x^3}$ can be rewritten using negative exponents as $f(x) = x^{-3}$ .
Apply power rule for differentiation: Apply the power rule for differentiation, which states that the derivative of $x^n$ is $n \cdot x^{n-1}$ . So, we will differentiate $f(x) = x^{-3}$ .
Perform the differentiation: Perform the differentiation: $f'(x) = -3x^{(-3 - 1)}$ . This follows directly from the power rule.
Simplify the expression: Simplify the expression by combining the exponents: $-3 - 1$ becomes $-4$ . So, the derivative $f'(x)$ is $-3x^{-4}$ .

More problems from Power rule II

Question

Find the derivative of

f(x)

\newline

f(x) = x^4

\newline

Write your answer as a constant times a power of

x

\newline

f'(x) =

Posted 9 months ago

Question

You pick a card at random, put it back, and then pick another card at random.

\newline

4

\newline

5

\newline

6

\newline

7

\newline

What is the probability of picking a number greater than

5

and then picking a

4

\newline

Write your answer as a percentage.

Posted 1 month ago

Question

Jeanne's bank account earns interest annually. The equation shows her starting balance of

\$500

and her balance at the end of three years,

\$546.36

r

did Jeanne earn interest?

\newline

546.36 = 500(1 + r)

Posted 1 month ago

Question

Jeanne's bank account earns interest annually. The equation shows her starting balance of

\$ 350

and her balance at the end of three years,

\$ 422.78

. At what rate r did Jeanne earn interest?

\newline

422.78=350(1+r)^{3}

Posted 1 month ago

Question

10\text{-kg}

bucket is lifted from the ground to a height of

14\, \text{m}

at a constant speed with a rope that weighs

0.6\, \text{kg/m}

. Initially the bucket contains

42\, \text{kg}

of water, but the water leaks at a constant rate and finishes draining just as the bucket reaches the

14\, \text{m}

level. How much work is done? (Use

9.8\, \text{m/s}^2

g

.) Show how to approximate the required work (in

\text{J}

) by a Riemann sum. (Let

x

be the height in meters above the ground. Enter

x_i^*

x_i

Posted 1 month ago

Question

10-\mathrm{kg}

bucket is lifted from the ground to a height of

14

m at a constant speed with a rope that weighs

0.6 \mathrm{~kg} / \mathrm{m}

. Initially the bucket contains

42

kg of water, but the water leaks at a constant rate and finishes draining just as the bucket reaches the

14-\mathrm{m}

level. How much work is done? (Use

9.8 \mathrm{~m} / \mathrm{s}^{2}

\newline

Show how to approximate the required work (in J) by a Riemann sum. (Let

x

be the height in meters above the ground. Enter

x_{i}^{* *}

x_{i}

\newline

\lim _{n \rightarrow \infty} \sum_{i=1}^{n}\left(\boxed{98+35.28 x_{i}}\right) \Delta x

\newline

Express the work (in J) as an integral in terms of

x

\newline

\int_{0}^{14}(98+35.28 x

\newline

Evaluate the integral (in J). (Round your answer to the nearest integer.)

Posted 1 month ago

Question

Solve the equation.

\newline

6 w^{7}=98,304

Posted 1 month ago

Question

10\text{-kg}

bucket is lifted from the ground to a height of

14\text{ m}

at a constant speed with a rope that weighs

0.6\text{ kg/m}

. Initially the bucket contains

42\text{ kg}

of water, but the water leaks at a constant rate and finishes draining just as the bucket reaches the

14\text{-m}

level. How much work is done? (Use

9.8\text{ m/s}^2

g

.) Show how to approximate the required work (in

\text{J}

) by a Riemann sum. (Let

x

be the height in meters above the ground. Enter

x_i^*

x_i

Posted 1 month ago

Question

\begin{array}{l}f(t)=\left\{\begin{array}{ll}-\frac{64}{t} & , \quad t=8 \\ 14-t & , \quad t=10 \\ t^{2}-3 t+2 & , \quad t \neq 8,10\end{array}\right. \\ f(2)=\square\end{array}

Posted 1 month ago

Question

\begin{array}{l}f(x)=\left\{\begin{array}{lll}\frac{x-6}{3} & , \quad x=-18 \\\frac{360}{x} & , \quad x=36 \\4 x+73 & , \quad x \neq-18,36\end{array}\right. \\f(-18)=\end{array}

Posted 1 month ago

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