How do you find the tangent line to #x=cos(y/4)# at #y=π#?

Answer 1

#-4sqrt2=dy/dx#

We have:

#x=cos(y/4)# Let's let #f(x)=y#.
#=>x=cos(f(x)/4)# Apply derivative on both sides.
#=>d/dx(x)=d/dx(cos(f(x)/4))#

Power rule:

#d/dx(x^n)=nx^(n-1)# where #n# is a constant.

Chain rule:

#d/dx(f(g(x)))=f'(g(x))*g'(x)#
#d/dx(cos(x))=-sin(x)#
#=>1*x^(1-1)=-sin(f(x)/4)*d/dx(f(x)/4)#
#=>x^(0)=-sin(f(x)/4)*1/4*d/dx(f(x))#
#=>4=-sin(f(x)/4)*f'(x)#
#=>4/-sin(f(x)/4)=f'(x)# Remember that #f(x)=y# and #f'(x)=dy/dx#
#=>4/-sin(y/4)=dy/dx#
When #y=pi...#
#=>4/-sin(pi/4)=dy/dx#
#=>4/(-sqrt2/2)=dy/dx#
#=>4*-2/sqrt2=dy/dx#
#=>-8/sqrt2=dy/dx#
#=>-4sqrt2=dy/dx#

That is the answer!

Sign up to view the whole answer

By signing up, you agree to our Terms of Service and Privacy Policy

Sign up with email
Answer 2

To find the tangent line to (x = \cos\left(\frac{y}{4}\right)) at (y = \pi), we first need to find the derivative of (x) with respect to (y), which will give us the slope of the tangent line. Then, we can evaluate this derivative at (y = \pi) to find the slope of the tangent line at that point. Finally, we can use the point-slope form of a line to write the equation of the tangent line.

First, we differentiate (x = \cos\left(\frac{y}{4}\right)) with respect to (y) using the chain rule:

[\frac{dx}{dy} = -\frac{1}{4}\sin\left(\frac{y}{4}\right)]

Now, we evaluate this derivative at (y = \pi):

[\frac{dx}{dy} \bigg|_{y=\pi} = -\frac{1}{4}\sin\left(\frac{\pi}{4}\right) = -\frac{1}{4}]

So, the slope of the tangent line at (y = \pi) is (-\frac{1}{4}).

Now, to find the equation of the tangent line, we use the point-slope form of a line:

[y - y_1 = m(x - x_1)]

Where (m) is the slope and ((x_1, y_1)) is the point on the curve.

At (y = \pi), (x = \cos\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}).

Substituting (m = -\frac{1}{4}), (x_1 = \frac{\sqrt{2}}{2}), and (y_1 = \pi) into the point-slope form, we get:

[y - \pi = -\frac{1}{4}\left(x - \frac{\sqrt{2}}{2}\right)]

This is the equation of the tangent line to (x = \cos\left(\frac{y}{4}\right)) at (y = \pi).

Sign up to view the whole answer

By signing up, you agree to our Terms of Service and Privacy Policy

Sign up with email
Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

Not the question you need?

Drag image here or click to upload

Or press Ctrl + V to paste
Answer Background
HIX Tutor
Solve ANY homework problem with a smart AI
  • 98% accuracy study help
  • Covers math, physics, chemistry, biology, and more
  • Step-by-step, in-depth guides
  • Readily available 24/7