Find the inverse of the function y = (square root x) + 4x , then solve for its 1st order derivative.

Find the inverse of the function y = (square root x) + 4x , then solve for its 1st order derivative.

switch variables and solve for y, and you get (after squaring)

16y^2 - (8x+1)y + x^2 = 0
y = [(8x+1)±√(16x+1)]/32

Pick the branch you want and take the derivative. Using the + branch,

y' = 1/4 (1 + 1/√(16x+1))

or, using implicit,

16x^2 - 8xy - x + y^2 = 0
32x - 8y - 8xy' + 2yy' = 0
(2y-8x)y' = 8y-32x

dy/dx = (4-16x)/(y-8x)

dx/dy = (√x+4x-8x)/(4-16x)
= (1/4)(√x-4x)/(1-4x)

Check my math and see whether these agree.

To find the inverse of the function y = √x + 4x, we need to follow these steps:

Step 1: Write the function in terms of x and y.
y = √x + 4x

Step 2: Swap x and y.
x = √y + 4y

Step 3: Solve for y.
Square both sides of the equation to eliminate the square root:
x^2 = (√y + 4y)^2
x^2 = y + 8y√y + 16y^2

Step 4: Rearrange the equation and collect like terms.
x^2 - y = 16y^2 + 8y√y
x^2 - y = y(16y + 8√y) + 16y^2

Step 5: Factor out y from the right side.
x^2 - y = y(16y + 8√y + 16y)

Step 6: Simplify and divide both sides by (16y + 8√y + 16y).
x^2 - y = y(16y + 8√y + 16y)/(16y + 8√y + 16y)
x^2 - y = y/(16 + 8√y + 16)

Step 7: Simplify the right side further.
x^2 - y = y/(32 + 8√y)
(x^2 - y)(32 + 8√y) = y
32x^2 + 8x^2√y - 32y - 8y√y = y
32x^2 + 8√y(x^2 - 4y) - 33y = 0

Step 8: Solve for √y.
8√y(x^2 - 4y) = 33y - 32x^2
√y(x^2 - 4y) = (33y - 32x^2)/8
√y = (33y - 32x^2)/(8(x^2 - 4y))
y = [(33y - 32x^2)/(8(x^2 - 4y))]^2
y = (33y - 32x^2)^2/(8^2(x^2 - 4y))
y = (1089y^2 - 2112yx^2 + 1024x^4)/(64(x^2 - 4y))

Step 9: Collect terms with y on one side.
y[1 + 64(x^2 - 4y)] = 1089y^2 - 2112yx^2 + 1024x^4
y + 64yx^2 -256y^2 = 1089y^2 - 2112yx^2 + 1024x^4
1089y^2 + (64x^2 - 2112y)x^2 + 256y^2 - y = 1024x^4

The equation obtained in Step 9 is the inverse of the original function y = √x + 4x. To find its first-order derivative, we differentiate this inverse function with respect to x. However, solving this equation is beyond the capacity of Explain Bot. You may use numerical methods or software tools to solve this equation and find the inverse function's first-order derivative.