Let P= (V^2)R / (R+r)^2
Calculate dP/dr, assuming that r is variable and R is constant.
(-2V^2R)/(R+r)^3
Oh, calculating derivatives? I guess it's time to put on my thinking hat! But don't worry, I promise I won't juggle the numbers. Let's get started!
To find dP/dr, we'll need to use the quotient rule. But first, let's rewrite our expression for P:
P = (V^2)R / (R+r)^2
Now, let's differentiate it step by step:
Step 1: Differentiate the numerator:
dP = d((V^2)R) / (R+r)^2
By using the product rule, we can differentiate (V^2)R as follows:
d((V^2)R) = 2VR*dV + (V^2)*dR
Step 2: Differentiate the denominator:
dP = (2VR*dV + (V^2)*dR) / (R+r)^2
To differentiate (R+r)^2, we'll use the chain rule:
d((R+r)^2) = 2(R+r)*d(r)
Step 3: Simplify the expression:
dP = (2VR*dV + (V^2)*dR) / (R+r)^2
Now, let's express d(r) in terms of dr, since r is the variable we're differentiating with respect to:
d(r) = dr
Step 4: Calculate dP/dr:
dP/dr = (2VR*dV + (V^2)*dR) / (R+r)^2
And there you have it! The derivative of P with respect to r, assuming R is constant, is dP/dr = (2VR*dV + (V^2)*dR) / (R+r)^2. Ta-da!
Remember, math can be a circus sometimes, but don't worry, I'm here to keep things entertaining.
To calculate dP/dr, we need to find the derivative of the equation P = (V^2)R / (R+r)^2 with respect to r, while assuming that R is constant.
To begin, let's rewrite the equation in a simplified form:
P = (V^2)R / (R+r)^2
P = (V^2)R * (R+r)^(-2)
Now, to find the derivative, we'll use the power rule and the chain rule:
Step 1: Apply the power rule to the numerator:
dP/dr = (V^2)R * (-2) * (R+r)^(-3)
Step 2: Apply the chain rule to the denominator by taking the derivative of R+r with respect to r, which is simply 1:
dP/dr = (V^2)R * (-2) * 1 * (R+r)^(-3)
Simplifying further:
dP/dr = -2(V^2)R / (R+r)^3
Therefore, the derivative of P with respect to r, dP/dr, equals -2(V^2)R / (R+r)^3.
P= (V^2)R / (R+r)^2
Assuming V,R are constants and use chain rule
dP/dr
= dP/d(R+r) * d(R+r)/dr
= d[(V^2)R/(R+r)^2]/r
=-2V^2R/(R+r)3
=-2V^2R/(R+r)3