Resources

Resources

Bytelearn - cat image with glasses

AI tutor

Welcome to Bytelearn!

Let’s check out your problem:

What is the value of
(d)/(dx)(x^(-3)) at
x=3 ?

What is the value of $\frac{d}{d x}\left(x^{-3}\right)$ at $x=3$ ?

View step-by-step help

View step-by-step help

Find values of derivatives using limits

Full solution

Q. What is the value of

\frac{d}{d x}\left(x^{-3}\right)

at

x=3

?

Identify Function & Operation: Identify the function and the operation needed. $\newline$ We need to find the derivative of the function $f(x) = x^{-3}$ with respect to $x$ and then evaluate this derivative at $x=3$ .
Apply Power Rule: Apply the power rule for differentiation. $\newline$ The power rule states that the derivative of $x^n$ with respect to $x$ is $n*x^{(n-1)}$ . Applying this rule to our function, we get: $\newline$ $f'(x) = \frac{d}{dx}(x^{-3}) = -3*x^{(-3-1)} = -3*x^{-4}.$
Simplify Derivative Expression: Simplify the expression for the derivative. $f'(x) = -3x^{-4}$ can be rewritten as $f'(x) = -\frac{3}{x^4}$ for easier evaluation.
Evaluate at $x=3$ : Evaluate the derivative at $x=3$ . $\newline$ Substitute $x$ with $3$ in the expression for $f'(x)$ : $\newline$ $f'(3) = -\frac{3}{3^4} = -\frac{3}{81}$ .
Simplify Result: Simplify the result. $\newline$ $-\frac{3}{81}$ simplifies to $-\frac{1}{27}$ .

More problems from Find values of derivatives using limits

Question

Consider the curve given by the equation

x y^{2}+5 x y=50

. It can be shown that

\frac{d y}{d x}=\frac{-y(y+5)}{x(2 y+5)} \text {. }

\newline

Write the equation of the vertical line that is tangent to the curve.

Posted 7 months ago

Question

The rate of change

\frac{d P}{d t}

of the number of algae in a tank is modeled by the following differential equation:

\newline

\frac{d P}{d t}=\frac{2317}{10614} P\left(1-\frac{P}{662}\right)

\newline

t=0

, the number of algae in the tank is

174

and is increasing at a rate of

28

algae per minute. At what value of

P

P(t)

growing the fastest?

\newline

Posted 9 months ago

Question

The rate of change

\frac{d P}{d t}

of the number of bacteria in a tank is modeled by the following differential equation:

\newline

\frac{d P}{d t}=\frac{2}{9849} P(598-P)

\newline

t=0

, the number of bacteria in the tank is

196

and is increasing at a rate of

16

bacteria per minute. At what value of

P

does the graph of

P(t)

have an inflection point?

\newline

Posted 9 months ago

Question

The rate of change

\frac{d P}{d t}

of the number of people infected by a disease is modeled by the following differential equation:

\newline

\frac{d P}{d t}=\frac{45}{125404} P(800-P)

\newline

t=0

, the number of people infected by the disease is

214

and is increasing at a rate of

45

people per hour. What is the limiting value for the total number of people infected by the disease as time increases?

\newline

Posted 9 months ago

Question

The exponential function

f

is graphed in the

x y

x

1, y

increases by a factor of

3

. Which of the following could be

f

\newline

1

\newline

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

\newline

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

\newline

f(x)=3^{x}+2

\newline

f(x)=2(3)^{x}

Posted 2 months ago

Question

Which of the following is a correct interpretation of the expression

\newline

-4+13

\newline

1

\newline

(A) The number that is

4

-13

on the number line

\newline

(B) The number that is

4

to the right of

-13

on the number line

\newline

(C) The number that is

13

-4

on the number line

\newline

(D) The number that is

13

to the right of

-4

on the number line

Posted 8 months ago

Question

f

f(x)=x^{3}+2 \sin (3 x-3)

. Use a calculator to write the equation of the line tangent to the graph of

f

x=0.5

. You should round all decimals to

3

\newline

Posted 9 months ago

Question

f

f(x)=x^{2}-2 x+3 \cos \left(x^{2}-x\right)

. Use a calculator to write the equation of the line tangent to the graph of

f

x=-2.5

. You should round all decimals to

3

\newline

Posted 9 months ago

Question

f

f(x)=x^{2}+x-2 \sin (2 x)

. Use a calculator to write the equation of the line tangent to the graph of

f

x=3

. You should round all decimals to

3

\newline

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

Related topics

Algebra - Order of Operations Algebra - Distributive Property `X` and `Y` Axes Geometry - Scalene Triangle Common Multiple Geometry - Quadrant