Resources

Resources

Bytelearn - cat image with glasses

AI tutor

Welcome to Bytelearn!

Let’s check out your problem:

8

5quad The variables
x and
y are such that
y=0 when
x=0 and

(x+1)y+(x+y+1)^(3)=1.
(a) Show that
(dy)/((d)x)=-(3)/(4) when
x=0.

$8$ $\newline$ $5 \quad$ The variables $x$ and $y$ are such that $y=0$ when $x=0$ and $\newline$ $(x+1) y+(x+y+1)^{3}=1 .$ $\newline$ (a) Show that $\frac{\mathrm{d} y}{\mathrm{~d} x}=-\frac{3}{4}$ when $x=0$ .

View step-by-step help

View step-by-step help

Evaluate expression when a complex numbers and a variable term is given

Full solution

Q.

8

\newline

5 \quad

The variables

x

and

y

are such that

y=0

when

x=0

and

\newline

(x+1) y+(x+y+1)^{3}=1 .

\newline

(a) Show that

\frac{\mathrm{d} y}{\mathrm{~d} x}=-\frac{3}{4}

when

x=0

.

Given Equation and Condition: Given the equation $(x+1)y + (x+y+1)^3 = 1$ and the condition $y=0$ when $x=0$ .
Implicit Differentiation: Differentiate both sides with respect to $x$ using implicit differentiation: $\frac{d}{dx}[(x+1)y] + \frac{d}{dx}[(x+y+1)^3] = \frac{d}{dx}[1]$ . For the first term, apply the product rule: $(x+1)\frac{dy}{dx} + y$ . For the second term, apply the chain rule: $3(x+y+1)^2 * (1 + \frac{dy}{dx})$ . Set the derivative of the constant ( $1$ ) to $0$ .
Substitute and Simplify: Substitute $y=0$ and $x=0$ into the differentiated equation: $(0+1)\frac{dy}{dx} + 0 + 3(0+0+1)^2 \cdot (1 + \frac{dy}{dx}) = 0$ . Simplify to get: $\frac{dy}{dx} + 3(1) \cdot (1 + \frac{dy}{dx}) = 0$ . This simplifies to $\frac{dy}{dx} + 3 + 3\frac{dy}{dx} = 0$ .
Combine and Solve: Combine like terms and solve for $\frac{dy}{dx}$ : $4\frac{dy}{dx} + 3 = 0$ . Subtract $3$ from both sides: $4\frac{dy}{dx} = -3$ . Divide by $4$ : $\frac{dy}{dx} = -\frac{3}{4}$ .

More problems from Evaluate expression when a complex numbers and a variable term is given

Question

\newline

8i + 6i

\newline

\newline

Write your answer in the form

a + bi

\newline

\underline{\hspace{1cm}}

\newline

Posted 2 months ago

Question

Find the complex conjugate of the given number.

\newline

4 + 10i

\newline

\newline

Write your answer in the form

a + bi

\newline

Posted 1 month ago

Question

\newline

\frac{{-4i}}{{-9i}}

\newline

\newline

Write your answer in the form

a + bi

. Reduce all fractions.

\newline

\_\_\_\_\_

\newline

Posted 1 month ago

Question

Combine the like terms to create an equivalent expression.

\newline

7k-k+19=\boxed{\phantom{0}}

Posted 4 months ago

Question

Apply the distributive property to factor out the greatest common factor.

\newline

9+12 n=

Posted 1 month ago

Question

Solve the equation.

\newline

\begin{aligned} & 12=n-9 \\ n= & \square \end{aligned}

Posted 4 months ago

Question

What is the amplitude of

y=-\sin (8 x-3)+5

\newline

Posted 2 months ago

Question

What is the period of

\newline

y=4 \sin (-2 x+7)-1 \text { ? }

\newline

Give an exact value.

\newline

Posted 1 month ago

Question

3 \cdot(3+20 i)=

\newline

Your answer should be a complex number in the form

a+b i

a

b

are real numbers.

Posted 3 months ago

Question

(35-23 i)+(13+25 i)=

\newline

Express your answer in the form

(a+b i)

Posted 1 month ago