find dy/d(theta)
theta=2 ,
if y=sin(pi/(square root(theta^2+5)))
Show steps thanks!!!
dy/dØ = cos(π/√(Ø^2+5) * (-1/2)(Ø^2 + 5)^(-3/2) ) * (2Ø)
=(-πØ/(Ø^2 + 5)^(3/2) * cos(π/√(Ø^2+5)
so when Ø = 2
dy/dØ = ...
You do the button - pushing.
Make sure to set your calculator to RAD
y = sin(π/√(θ^2+5))
weird, but just apply the rules
y = sun(u)
y' = cos(u) u'
u = π/v
u' = -π/v^2 v'
v = √(θ^2+5)
v' = θ/√(θ^2+5)
put it all together and it spells MOTHER. Or, more mathematically,
πθ*cos(π/√(θ^2+5)) / (θ^2+5)^(3/2)
y = sin [ pi*(t^2+5)^-.5 ]
y' = pi [cos pi*(t^2+5)^-.5] * [-.5(2t)](t^2+5)^-1.5
if t = 2
y' = pi {cos pi/ 3} *[-2] [ 1/27 ]
= -(2pi/27) cos (pi/3)
but cos pi/3 = 1/2
so
-pi/27
CHECK MY ARITHMETIC !!!!
=
To find dy/d(theta), we can use the chain rule. The chain rule states that if we have a composition of functions, u = f(g(theta)), then the derivative of u with respect to theta, du/d(theta), is given by du/d(theta) = (du/dg) * (dg/d(theta)).
In this case, we have y = sin(pi/(sqrt(theta^2+5))). Let's break it down and find the derivative step by step:
Step 1: Find the derivative of the inner function.
Let's find dy/dg first, where g(theta) = sqrt(theta^2 + 5).
To differentiate g(theta), we can use the power rule and chain rule:
dg/d(theta) = (1/2) * (1/sqrt(theta^2+5)) * (2*theta)
= (theta / sqrt(theta^2 + 5))
Step 2: Find the derivative of the outer function.
The derivative of sin(u) with respect to u is cos(u). In this case, u = pi/(sqrt(theta^2+5)).
So, dy/du = cos(pi/(sqrt(theta^2+5)))
Step 3: Apply the chain rule to find dy/d(theta).
Using the chain rule, we have:
dy/d(theta) = dy/du * du/d(theta)
= cos(pi/(sqrt(theta^2+5))) * (theta / sqrt(theta^2 + 5))
Step 4: Substitute the value for theta.
Since theta = 2, we can substitute it into the expression:
dy/d(theta) = cos(pi/(sqrt(2^2+5))) * (2 / sqrt(2^2 + 5))
= cos(pi/(sqrt(4 + 5))) * (2 / sqrt(4 + 5))
= cos(pi/(sqrt(9))) * (2 / sqrt(9))
= cos(pi/3) * (2/3)
= (1/2) * (2/3)
= 1/3
Therefore, dy/d(theta) = 1/3 when theta = 2.