Resourcesdown chevron
Testimonials
Plans
Sign in
Sign up
Resourcesright arrow
Testimonials
Plans
Bytelearn - cat image with glassesAI tutor

Welcome to Bytelearn!

Let’s check out your problem:

A curve is defined by the parametric equations x(t)=-6sin(-9t) and y(t)=4cos(4t). Find (dy)/(dx). Answer:

A curve is defined by the parametric equations x(t)=−6sin⁡(−9t) x(t)=-6 \sin (-9 t) and y(t)=4cos⁡(4t) y(t)=4 \cos (4 t) . Find dydx \frac{d y}{d x} .\newlineAnswer:

View step-by-step helpView step-by-step help
Homeright chevron icon
Math Problemsright chevron icon
Algebra 2right chevron icon
Write equations of cosine functions using propertiesright chevron icon

Full solution

Q. A curve is defined by the parametric equations x(t)=−6sin⁡(−9t) x(t)=-6 \sin (-9 t) and y(t)=4cos⁡(4t) y(t)=4 \cos (4 t) . Find dydx \frac{d y}{d x} .\newlineAnswer:
  1. Find dxdt\frac{dx}{dt}: To find dydx\frac{dy}{dx} for parametric equations, we need to find dydt\frac{dy}{dt} and dxdt\frac{dx}{dt} separately and then divide dydt\frac{dy}{dt} by dxdt\frac{dx}{dt}.
  2. Find dydt\frac{dy}{dt}: First, we find dxdt\frac{dx}{dt}. The derivative of x(t)=−6sin⁡(−9t)x(t) = -6\sin(-9t) with respect to tt is dxdt=−6×−9×cos⁡(−9t)=54cos⁡(−9t)\frac{dx}{dt} = -6 \times -9 \times \cos(-9t) = 54\cos(-9t), because the derivative of sin⁡(u)\sin(u) with respect to uu is cos⁡(u)\cos(u) and we apply the chain rule for the derivative of −9t-9t.
  3. Divide to find dydx\frac{dy}{dx}: Next, we find dydt\frac{dy}{dt}. The derivative of y(t)=4cos⁡(4t)y(t) = 4\cos(4t) with respect to tt is dydt=−4×4×sin⁡(4t)=−16sin⁡(4t)\frac{dy}{dt} = -4 \times 4 \times \sin(4t) = -16\sin(4t), because the derivative of cos⁡(u)\cos(u) with respect to uu is −sin⁡(u)-\sin(u) and we apply the chain rule for the derivative of 4t4t.
  4. Simplify the expression: Now we divide dydt\frac{dy}{dt} by dxdt\frac{dx}{dt} to find dydx\frac{dy}{dx}. So, dydx=−16sin⁡(4t)54cos⁡(−9t)\frac{dy}{dx} = \frac{-16\sin(4t)}{54\cos(-9t)}.
  5. Simplify the expression: Now we divide dydt\frac{dy}{dt} by dxdt\frac{dx}{dt} to find dydx\frac{dy}{dx}. So, dydx=−16sin⁡(4t)54cos⁡(−9t)\frac{dy}{dx} = \frac{-16\sin(4t)}{54\cos(-9t)}.We can simplify the expression by dividing both the numerator and the denominator by 22, which gives us dydx=−8sin⁡(4t)27cos⁡(−9t)\frac{dy}{dx} = \frac{-8\sin(4t)}{27\cos(-9t)}.

More problems from Write equations of cosine functions using properties

Question
Find an equation for a sinusoidal function that has period 2π2\pi, amplitude 11, and contains the point (π,−1)\left(\pi, -1\right). \newlineWrite your answer in the form f(x)=Acos⁡(Bx+C)+Df(x)=A\cos(Bx+C)+D, where AA, BB, CC, and DD are real numbers.\newline f(x)=f(x)=
Get tutor helpright-arrow

Posted 7 months ago

Question
Find an equation for a sinusoidal function that has period 2π2\pi, amplitude 11, and contains the point (−π2,−1)\left(-\frac{\pi}{2},-1\right). Write your answer in the form f(x)=Asin⁡(Bx+C)+Df(x) = A\sin(Bx + C) + D, where AA, BB, CC, and DD are real numbers. `f(x)` =______
Get tutor helpright-arrow

Posted 9 months ago

Question
Find an equation for a sinusoidal function that has period 2π2\pi, amplitude 11, and contains the point (0,−1)(0,-1). \newlineWrite your answer in the form f(x)=Acos⁡(Bx+C)+Df(x) = A\cos(Bx + C) + D, where AA, BB, CC, and DD are real numbers. \newlinef(x)=f(x) = _____
Get tutor helpright-arrow

Posted 8 months ago

Question
Franklin Middle School is having a canned food drive. In Mr. Lao's class, Mr. Lao brought 1010 cans of food, and each student brought 22 cans. In Ms. Roger's class, Ms. Rogers did not bring any cans, but each student brought 33 cans of food. Write two equations in slope-intercept form to represent the number of cans of food for each class yy in terms of the number of students xx.
Get tutor helpright-arrow

Posted 8 months ago

Question
Franklin Middle School is having a canned food drive. In Mr. Lao's class, Mr. Lao brought 1010 cans of food, and each student brought 22 cans. In Ms. Roger's class, Ms. Rogers did not bring any cans, but each student brought 33 cans of food. Write two equations in slope-intercept form to represent the number of cans of food for each class yy in terms of the number of students xx.
Get tutor helpright-arrow

Posted 8 months ago

Question
Kris is wrapping Christmas lights around the railing that runs around three sides of his square porch If one side of his porch is 1212 feet long, and each strand of his Christmes lights will wrap around a 7070-inch length of the railing, what is the minimum number of strands Kris needs to completely wrap the porch railing?\newlineA) 66\newlineB) 77\newlineC) 88\newlineD) 99
Get tutor helpright-arrow

Posted 8 months ago

Question
Kris is wrapping Christmas lights around the railing that runs around three sides of his square porch If one side of his porch is 1212 feet long, and each strand of his Christmas lights will wrap around a 7070-inch length of the railing, what is the minimum number of strands Kris needs to completely wrap the porch railing?\newlineA) 66\newlineB) 77\newlineC) 88\newlineD) 99
Get tutor helpright-arrow

Posted 8 months ago

Question
For a given input value b b , the function f f outputs a value a a to satisfy the following equation.\newline4a+7b=−52 4a+7b=-52 \newlineWrite a formula for f(b) f(b) in terms of b b .\newlinef(b)=□ f(b)=\square
Get tutor helpright-arrow

Posted 6 months ago

Question
For a given input value xx, the function gg outputs a value yy to satisfy the following equation.\newline−4x−6=−5y+2-4x-6=-5y+2\newlineWrite a formula for g(x)g(x) in terms of xx.\newlineg(x)=□g(x)=\square
Get tutor helpright-arrow

Posted 10 months ago

Question
For a given input value aa, the function ff outputs a value bb to satisfy the following equation.\newline−3a+6b=a+4b-3a+6b=a+4b\newlineWrite a formula for f(a)f(a) in terms of aa.\newlinef(a)=□f(a)=\square
Get tutor helpright-arrow

Posted 10 months ago

View all (68)

Related topics

Algebra - Order of OperationsAlgebra - Distributive Property`X` and `Y` AxesGeometry - Scalene TriangleCommon MultipleGeometry - Quadrant