Resources

Resources

Bytelearn - cat image with glasses

AI tutor

Welcome to Bytelearn!

Let’s check out your problem:

If
y=5y^(3)+2x^(3) then find
(dy)/(dx) in terms of
x and
y.
Answer:
(dy)/(dx)=

If $y=5 y^{3}+2 x^{3}$ then find $\frac{d y}{d x}$ in terms of $x$ and $y$ . $\newline$ Answer: $\frac{d y}{d x}=$

View step-by-step help

View step-by-step help

Evaluate an exponential function

Full solution

Q. If

y=5 y^{3}+2 x^{3}

then find

\frac{d y}{d x}

in terms of

x

and

y

.

\newline

Answer:

\frac{d y}{d x}=

Differentiation Process: To find the derivative of $y$ with respect to $x$ , we need to differentiate both sides of the equation with respect to $x$ . The equation is $y = 5y^3 + 2x^3$ .
Derivative of $y$ : Differentiate the left side of the equation with respect to $x$ . The derivative of $y$ with respect to $x$ is simply $\frac{dy}{dx}$ .
Derivative of Right Side: Differentiate the right side of the equation with respect to $x$ . The term $5y^3$ is treated as a function of $x$ , so we apply the chain rule. The derivative of $5y^3$ with respect to $x$ is $15y^2 \cdot \frac{dy}{dx}$ . The derivative of $2x^3$ with respect to $x$ is $6x^2$ .
Combining Derivatives: Combine the derivatives to form the equation: $\frac{dy}{dx} = 15y^2 \cdot \left(\frac{dy}{dx}\right) + 6x^2$ .
Isolating $\frac{dy}{dx}$ : To solve for $\frac{dy}{dx}$ , we need to move all terms involving $\frac{dy}{dx}$ to one side of the equation. Subtract $15y^2 \times \frac{dy}{dx}$ from both sides to get: $\frac{dy}{dx} - 15y^2 \times \frac{dy}{dx} = 6x^2$ .
Factoring out $\frac{dy}{dx}$ : Factor out $\frac{dy}{dx}$ on the left side of the equation to get: $\frac{dy}{dx} \times (1 - 15y^2) = 6x^2$ .
Final Derivative Solution: Divide both sides of the equation by $(1 - 15y^2)$ to solve for $\frac{dy}{dx}$ . This gives us: $\frac{dy}{dx} = \frac{6x^2}{(1 - 15y^2)}$ .

More problems from Evaluate an exponential function

Question

Use the following function rule to find

f(2)

\newline

f(x) = 11 \cdot (9)^x + 8

\newline

f(2) =

Posted 2 months ago

Question

Raymond works for a pharmaceutical company that is testing the effectiveness of a new medication. He gave one of the study participants

400

milligrams of the medication, and he knows the amount remaining in the participant's body will decrease by a factor of

1/3

\newline

Write an exponential equation in the form

y = a(b)^x

that can model the amount of medication,

y

, remaining in the participant's body after

x

\newline

Use whole numbers, decimals, or simplified fractions for the values of

a

b

\newline

y =

\newline

Posted 2 months ago

Question

100

bacteria doubles in size every

60

hours. What will the population be

180

hours from now?

\newline

Posted 2 months ago

Question

Charlotte opened a savings account and deposited

\$800.00

as principal. The account earns

8\%

interest, compounded annually. What is the balance after

10

\newline

Use the formula

A = P(1 + \frac{r}{n})^{nt}

A

is the balance (final amount),

P

is the principal (starting amount),

r

is the interest rate expressed as a decimal,

n

is the number of times per year that the interest is compounded, and

t

is the time in years.

\newline

Round your answer to the nearest cent.

\newline

\$

\newline

Posted 2 months ago

Question

Bob rented a car for a flat fee of `$150` and an additional `$10` per day. Write a linear function that represents the total cost,

f(x)

x

Posted 3 months ago

Question

Sally purchased a laptop for `$800` and has to pay a service fee of `$20` for each day until it is delivered. Write a linear function to represent the total cost,

h(x)

x

Posted 3 months ago

Question

\$1,000

in a savings account with a daily interest rate of

0.1\%

. Write an exponential function to represent the total amount,

s(x)

, in his account after

x

Posted 3 months ago

Question

A=2450+740 s

\newline

The total amount,

A

, in thousands of dollars, of scholarship money given out by a college throughout its history at a time of

s

semesters after the summer of the year

2000

is given by the equation. What was the total amount of scholarship money, in thousands of dollars, given out by the college up to the summer of the year

2000

Posted 1 month ago

Question

T=235+980 d

\newline

T

, in newton-meters, on a diving board with a

100

kilogram weight placed

d

meters from the diving board's fulcrum is given by the equation. How much torque in newton-meters is on the diving board when the weight is placed at the fulcrum?

Posted 2 months ago

Question

R=42-0.7 t

\newline

Quinn returned home one summer's day to find it extremely hot. He turned the air conditioner on, and the room's temperature began decreasing at a constant rate. The given equation shown gives the room's temperature,

R

, in degrees Celsius,

t

minutes after Quinn turned on the air conditioner. What was the room's temperature, in degrees Celsius, when Quinn returned home?

Posted 1 month ago