Respuesta :
Answer:
dy/dx = -1/√(1 - x²)
For 0 < y < π
Step-by-step explanation:
Given the function cos y = x
-siny dy = dx
-siny dy/dx = 1
dy/dx = -1/siny (equation 1)
But cos²y + sin²y = 1
=> sin²y = 1 - cos²y
=> siny = √(1 - cos²y) (equation 2)
Again, we know that
cosy = x
=> cos²y = x² (equation 3)
Using (equation 3) in (equation 2), we have
siny = √(1 - x²) (equation 4)
Finally, using (equation 4) in (equation 1), we have
dy/dx = -1/√(1 - x²)
The largest interval is when
√(1 - x²) = 0
=> 1 - x² = 0
=> x² = 1
=> x = ±1
So, the interval is
-1 < x < 1
arccos(1) < y < arxcos(-1)
= 0 < y < π
The required value is,
[tex]\frac{dy}{dx}=-\frac{1}{siny}[/tex]
Differentiation:
Differentiation, in calculus, can be applied to measure the function per unit change in the independent variable.
It is given that,
[tex]cos y=x[/tex]
Now, differentiating the function with respect to [tex]x[/tex].
[tex]-siny \frac{dy}{dx} =1\\\frac{dy}{dx}=-\frac{1}{siny}[/tex]
We have to find the largest interval so that [tex]y[/tex] is a differentiable function [tex]x[/tex]Then,
[tex]\frac{dy}{dx}=-\frac{1}{siny}[/tex]
The function[tex]\frac{1}{siny}[/tex] is defined only if [tex]siny\neq 0[/tex]
Therefore [tex]y\neq n \pi[/tex] where [tex]n \in \mathbb{Z}[/tex]
Then,[tex]-n \pi[/tex]
Hence, the function is a differentiable function of [tex]x[/tex] if [tex]-n \pi[/tex]
Learn more about the topic differentiation:
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