A. find y' by implicit differentiation.
B. solve th equation explicitly for y and differentiate to get y' in terms of x.
cos (x) + √y = 5
a)
- sin(x) + (1/2)y^(-1/2)dy/dx = 0
-(1/4)(1/√y)dy/dx = sin(x)
dy/dx = -4√y sin(x)
b)
√y = 5 - cosx
y = 25 - 10cosx + cos^2 x
dy/dx = -10cosx + 2(cosx)(-sinx)
= -2cosx(5cos(x) + 2sin(x))
A. Oh, no, not implicit differentiation! It's like trying to find your way through a maze blindfolded. But fear not, I'll guide you through it. Let's find y' in terms of x for the equation cos(x) + √y = 5.
To find y', we'll need to differentiate both sides of the equation with respect to x. So, let's get our differentiation cap on!
Differentiating cos(x) with respect to x gives us -sin(x).
Differentiating √y with respect to x requires a little trickery. We can express √y as y^(1/2), and then apply the chain rule. The derivative of y^(1/2) with respect to y is (1/2)y^(-1/2), and then we multiply it by y' to get the total derivative.
So, we have -sin(x) + (1/2)y^(-1/2) * y' = 0.
Now, let's solve this equation for y', which is what we're ultimately after. We need to isolate y' on one side, so let's move the -sin(x) term to the other side:
(1/2)y^(-1/2) * y' = sin(x).
To get y' by itself, we multiply both sides by 2 and divide by y^(-1/2):
y' = 2sin(x) * y^(-1/2).
And there you have it! The derivative of y with respect to x, obtained through the magic of implicit differentiation.
B. If you'd rather avoid the road of implicit differentiation and solve the equation explicitly for y, we can give that a try too.
cos(x) + √y = 5
First, let's move the cos(x) term to the other side:
√y = 5 - cos(x)
To get rid of the square root, we'll square both sides:
y = (5 - cos(x))^2
Now, to find y' in terms of x, we'll differentiate y with respect to x. But be aware that when squaring both sides, we must also use the chain rule.
Differentiating y = (5 - cos(x))^2 with respect to x, we get:
y' = 2(5 - cos(x))(0 - sin(x))
Simplifying that expression:
y' = -2(5 - cos(x))sin(x)
And there you go! The derivative of y with respect to x, obtained through explicit differentiation.
A. To find y' by implicit differentiation, we'll differentiate each term with respect to x, treating y as a function of x.
Differentiating cos(x) with respect to x gives -sin(x).
Differentiating √y with respect to x requires the chain rule. Let's denote √y as u, so u = √y. Then, we can rewrite the equation as u = √y = 5 - cos(x).
Differentiating both sides using the chain rule, we get:
d/dx(u) = d/dx(5 - cos(x))
du/dx = -sin(x)
Now, we have du/dx = d/dx(√y) = (1/2)(1/√y) dy/dx = (1/2)(1/√y) y'.
Therefore, we can solve for y':
(1/2)(1/√y) y' = -sin(x)
y' = -2sin(x)√y
B. To solve the equation explicitly for y and differentiate to get y' in terms of x, we'll manipulate the equation to isolate y and then differentiate explicitly.
Start with the equation:
cos(x) + √y = 5
Subtract cos(x) from both sides:
√y = 5 - cos(x)
Square both sides to eliminate the square root:
y = (5 - cos(x))^2
Differentiate both sides with respect to x:
dy/dx = 2(5 - cos(x))(-sin(x))
Therefore, y' in terms of x is:
y' = 2(5 - cos(x))(-sin(x))
To find y' using implicit differentiation, follow these steps:
Step 1: Start by differentiating both sides of the equation with respect to x. Each term will be differentiated separately.
d/dx [cos(x) + √y] = d/dx [5]
Step 2: The derivative of cos(x) with respect to x is -sin(x). To differentiate √y with respect to x, we need to use the chain rule. The derivative of √y with respect to y is 1/(2√y), so we multiply it by the derivative of y with respect to x (y').
-sin(x) + (1/(2√y)) * y' = 0
Step 3: Now, we can isolate y' to one side of the equation. Move -sin(x) to the other side.
(1/(2√y)) * y' = sin(x)
Step 4: Multiply both sides by (2√y) to get y' by itself.
y' = 2√y * sin(x)
So, the derivative of y with respect to x, y', is equal to 2√y * sin(x).
Alternatively, to solve the equation explicitly for y and then differentiate to find y' in terms of x, follow these steps:
Step 1: Start by isolating √y on one side of the equation.
√y = 5 - cos(x)
Step 2: Square both sides of the equation to remove the square root.
y = (5 - cos(x))^2
Step 3: Differentiate y with respect to x using the power rule and chain rule.
y' = 2(5 - cos(x)) * (-sin(x))
Simplifying further,
y' = -2(5 - cos(x)) * sin(x)
So, the derivative of y with respect to x, y', is equal to -2(5 - cos(x)) * sin(x).