C = 15*pi*(2*D*h+(3/4)*D²) + L* (94.25*D*h+35.34*D^2)/(pi*D^2*h/4)
first dC/dD?
second d^2C/dD^2?
first dC/dh?
second d^2C/dh^2
To find the first derivative of C with respect to D (dC/dD), you need to use the power rule and the product rule.
From the given equation:
C = 15*pi*(2*D*h+(3/4)*D²) + L*(94.25*D*h+35.34*D^2)/(pi*D^2*h/4)
Step 1: Expand the terms within the parentheses:
C = 15*pi*(2*D*h+ (3/4)*D²) + L*(94.25*D*h+35.34*D^2)/(pi*D^2*h/4)
= 30*pi*D*h+ (45/4)*pi*D² + L*(94.25*D*h+35.34*D^2)/(pi*D^2*h/4)
Step 2: Differentiate each term separately:
dC/dD = 30*pi*h + (45/2)*pi*D + L*(94.25*h+70.68*D)/(pi*D^2*h/4)
Therefore, the first derivative of C with respect to D (dC/dD) is:
dC/dD = 30*pi*h + (45/2)*pi*D + L*(94.25*h+70.68*D)/(pi*D^2*h/4)
Now let's move on to finding the second derivative (d²C/dD²).
To find the second derivative, you need to differentiate the first derivative, which is dC/dD, with respect to D again.
Step 1: Differentiate the expression dC/dD with respect to D:
d²C/dD² = (45/2)*pi + L*(70.68)/(pi*D^2*h/4)*(-2)*(94.25*h+70.68*D)/(pi*D^2*h/4)
= (45/2)*pi + L*(70.68)*(94.25*h+70.68*D)/(pi*D^2*h/4)^2
Therefore, the second derivative of C with respect to D (d²C/dD²) is:
d²C/dD² = (45/2)*pi + L*(70.68)*(94.25*h+70.68*D)/(pi*D^2*h/4)^2
Now let's move on to finding the first and second derivatives with respect to h.
To find the first derivative of C with respect to h (dC/dh), you need to differentiate the expression C with respect to h while considering D as a constant.
Step 1: Differentiate the expression C with respect to h:
dC/dh = 30*pi*D + L*(94.25*D + 70.68*D^2)/(pi*D^2*h/4)
Therefore, the first derivative of C with respect to h (dC/dh) is:
dC/dh = 30*pi*D + L*(94.25*D + 70.68*D^2)/(pi*D^2*h/4)
To find the second derivative of C with respect to h (d²C/dh²), you need to differentiate the first derivative (dC/dh) with respect to h again.
Step 1: Differentiate the expression dC/dh with respect to h:
d²C/dh² = L*(94.25*D + 70.68*D^2)/(-pi*D^2*h^2/4)
Therefore, the second derivative of C with respect to h (d²C/dh²) is:
d²C/dh² = L*(94.25*D + 70.68*D^2)/(-pi*D^2*h^2/4)