Resources

Resources

Bytelearn - cat image with glasses

AI tutor

Welcome to Bytelearn!

Let’s check out your problem:

Using implicit differentiation, find
(dy)/(dx).

sqrt(xy)=-4+xy^(3)

Using implicit differentiation, find $\frac{d y}{d x}$ . $\newline$ $\sqrt{x y}=-4+x y^{3}$

View step-by-step help

View step-by-step help

Find trigonometric ratios of special angles

Full solution

Q. Using implicit differentiation, find

\frac{d y}{d x}

.

\newline

\sqrt{x y}=-4+x y^{3}

Apply Chain and Product Rule: First, we need to differentiate both sides of the equation with respect to $x$ . The equation is $\sqrt{xy} = -4 + xy^3$ . We will apply the chain rule and product rule where necessary.
Differentiate Left Side: Differentiate the left side with respect to $x$ . The left side is $\sqrt{xy}$ , which is $(xy)^{\frac{1}{2}}$ . Using the chain rule and product rule, we get $\frac{1}{2}(xy)^{-\frac{1}{2}} \times (x\frac{dy}{dx} + y)$ .
Differentiate Right Side: Differentiate the right side with respect to $x$ . The right side is $-4 + xy^3$ . The derivative of $-4$ is $0$ , and using the product rule for $xy^3$ , we get $y^3 + 3xy^2\frac{dy}{dx}$ .
Equate Derivatives: Now we equate the derivatives from both sides: $(\frac{1}{2})(xy)^{-\frac{1}{2}} * (x\frac{dy}{dx} + y) = 0 + y^3 + 3xy^2\frac{dy}{dx}$ .
Simplify Equation: Simplify the equation by multiplying both sides by $2(xy)^{1/2}$ to get rid of the fraction and the square root: $x\frac{dy}{dx} + y = 2(xy)^{1/2}(y^3 + 3xy^2\frac{dy}{dx})$ .
Expand Right Side: Expand the right side of the equation: $x\frac{dy}{dx} + y = 2y^3(xy)^{\frac{1}{2}} + 6x^2y^2\frac{dy}{dx}$ .
Group Terms: Group the terms with $\frac{dy}{dx}$ on one side and the rest on the other side: $x\frac{dy}{dx} - 6x^2y^2\frac{dy}{dx} = 2y^3\sqrt{xy} - y$ .
Factor Out: Factor out $(\frac{dy}{dx})$ on the left side: $(\frac{dy}{dx})(x - 6x^2y^2) = 2y^3(xy)^{\frac{1}{2}} - y$ .
Solve for $\frac{dy}{dx}$ : Solve for $\frac{dy}{dx}$ by dividing both sides by $(x - 6x^2y^2)$ : $\frac{dy}{dx} = \frac{2y^3(xy)^{\frac{1}{2}} - y}{x - 6x^2y^2}$ .
Final Simplification: Simplify the expression for $\frac{dy}{dx}$ if possible. However, in this case, the expression is already in its simplest form.

More problems from Find trigonometric ratios of special angles

Question

Find all solutions with

0^\circ \leq \theta \leq 180^\circ

. Give the exact answer(s) in simplest form. If there are multiple answers, separate them with commas.

\newline

\cos(\theta) = -1

\newline

\underline{\hspace{2em}}^\circ

Posted 1 month ago

Question

Kimi's school is

15

kilometers west of her house and

8

kilometers south of her friend Franklin's house. Every day, Kimi bicycles from her house to her school. After school, she bicycles from her school to Franklin's house. Before dinner, she bicycles home on a bike path that goes straight from Franklin's house to her own house. How far does Kimi bicycle each day?

\newline

\_\_\_\_\_

Posted 1 month ago

Question

Evaluate. Write your answer as an integer or as a decimal rounded to the nearest hundredth.

\newline

\cos 35^\circ =

Posted 1 month ago

Question

Find all solutions with

-90^\circ \leq \theta \leq 90^\circ

. Give the exact answer(s) in simplest form. If there are multiple answers, separate them with commas.

\newline

\csc(\theta) = -1

\newline

\_\_\_\_\,^\circ

Posted 1 month ago

Question

Evaluate. Write your answer in simplified, rationalized form. Do not round.

\newline

\cot 30^\circ =

Posted 1 month ago

Question

The terminal side of an angle

\theta

in standard position intersects the unit circle at

(\frac{84}{85}, \frac{13}{85})

\cos(\theta)

\newline

Write your answer in simplified, rationalized form.

\newline

Posted 1 month ago

Question

-90^\circ < \theta < 90^\circ

. Find the value of

\theta

\newline

\tan(\theta) = 0

\newline

Write your answer in simplified, rationalized form. Do not round.

\newline

\theta =

^\circ

\newline

Posted 2 months ago

Question

Evaluate. Write your answer in simplified, rationalized form. Do not round.

\newline

\csc 60^\circ =

Posted 1 month ago

Question

\theta

is an acute angle. Find the value of

\theta

\newline

\sec(\theta) = \sqrt{2}

\newline

\theta =

\degree

Posted 1 month ago

Question

Evaluate. Write your answer in simplified, rationalized form. Do not round.

\newline

\cos 90^\circ =

Posted 2 months ago