Area of rectangle = xy
= x�ã(4r^2 - x^2)
Take the derivative. I am having trouble with the algebra.
x *(4r^2 - x^2)^-1/2
Sorry. I'm having trouble with the font. Here is the correct equation.
Area of rectangle = xy
=x *(4r^2 - x^2)^1/2
Using the product rule
d(area)/dx = x(1/2)(4r^2-x^2)^(-1/2)(-2x) + (4r^2-x^2)^(1/2)
= -x^2(4r^2-x^2)^(-1/2) + (4r^2-x^2)^(1/2)
= (4r^2-x^2)^(-1/2)[-x^2 + 4r^2-x^2]
= (4r^2 - 2x^2)(4r^2-x^2)^(-1/2)
you will probably set this equal to zero
to solve
4r^2 - 2x^2 = 0
etc
Thank you. It makes so much sense now.
To find the derivative of the expression, you can use the product rule and the chain rule. Let's break down the steps:
Step 1: Apply the product rule
The product rule states that for two functions u(x) and v(x), (u * v)' = u' * v + u * v'. In this case, let u(x) = x and v(x) = (4r^2 - x^2). Applying the product rule:
d/dx (x * (4r^2 - x^2)) = (d/dx(x)) * (4r^2 - x^2) + x * (d/dx(4r^2 - x^2))
Step 2: Simplify the derivatives
The derivative of 'x' with respect to 'x' is simply 1, and the derivative of '4r^2 - x^2' with respect to 'x' can be found using the chain rule. Let's break it down further:
d/dx (4r^2 - x^2) = d/dx(4r^2) - d/dx(x^2)
The derivative of a constant (like 4r^2) with respect to 'x' is 0. Then, applying the chain rule on 'x^2', we get:
d/dx (4r^2 - x^2) = 0 - d/dx(x^2) = -2x
Step 3: Final derivative
Substituting the simplified derivatives back into the product rule equation:
(1) * (4r^2 - x^2) + x * (-2x) = 4r^2 - x^2 - 2x^2 = 4r^2 - 3x^2
Therefore, the derivative of xy with respect to x is 4r^2 - 3x^2.
I hope this explanation helps you understand the algebraic steps involved!