Derivative:3y^3-4x^(2y)+xy=-5
The derivative of y is y'(x):
(d/dx(x)) y+y'(x) 9 y^2+x y'(x)-8 x^(2 y) ((d/dx(log(x))) y+log(x) y'(x)) = d/dx(-5)
The derivative of x is 1:
1 y+x y'(x)+9 y^2 y'(x)-8 x^(2 y) ((d/dx(log(x))) y+log(x) y'(x)) = d/dx(-5)
The derivative of log(x) is 1/x:
y+x y'(x)+9 y^2 y'(x)-8 x^(2 y) (1/x y+log(x) y'(x)) = d/dx(-5)
The derivative of -5 is zero:
y+x y'(x)+9 y^2 y'(x)-8 x^(2 y) (y/x+log(x) y'(x)) = 0
Expand the left hand side:
y-8 x^(-1+2 y) y+x y'(x)-8 x^(2 y) log(x) y'(x)+9 y^2 y'(x) = 0
Subtract y-8 y x^(2 y-1) from both sides:
x y'(x)-8 x^(2 y) log(x) y'(x)+9 y^2 y'(x) = -y+8 x^(-1+2 y) y
Collect the left hand side in terms of y'(x):
(x-8 x^(2 y) log(x)+9 y^2) y'(x) = -y+8 x^(-1+2 y) y
Divide both sides by -8 x^(2 y) log(x)+x+9 y^2:
Answer: |y'(x) = (-y+8 x^(-1+2 y) y)/(x-8 x^(2 y) log(x)+9 y^2)
Derivative:y^3+sinhxy^2=3/2
Find the derivative of the following via implicit differentiation:
d/dx(sinh^2(x y)+y^3) = d/dx(3/2)
Differentiate the sum term by term:
d/dx(sinh^2(x y))+d/dx(y^3) = d/dx(3/2)
Using the chain rule, d/dx(sinh^2(x y)) = ( du^2)/( du) ( du)/( dx), where u = sinh(x y) and ( d)/( du)(u^2) = 2 u:
d/dx(y^3)+2 d/dx(sinh(x y)) sinh(x y) = d/dx(3/2)
Using the chain rule, d/dx(sinh(x y)) = ( dsinh(u))/( du) ( du)/( dx), where u = x y and ( d)/( du)(sinh(u)) = cosh(u):
d/dx(y^3)+cosh(x y) d/dx(x y) 2 sinh(x y) = d/dx(3/2)
Using the chain rule, d/dx(y^3) = ( du^3)/( du) ( du)/( dx), where u = y and ( d)/( du)(u^3) = 3 u^2:
2 cosh(x y) (d/dx(x y)) sinh(x y)+3 d/dx(y) y^2 = d/dx(3/2)
The derivative of y is y'(x):
2 cosh(x y) (d/dx(x y)) sinh(x y)+y'(x) 3 y^2 = d/dx(3/2)
Use the product rule, d/dx(u v) = v ( du)/( dx)+u ( dv)/( dx), where u = x and v = y:
x d/dx(y)+d/dx(x) y 2 cosh(x y) sinh(x y)+3 y^2 y'(x) = d/dx(3/2)
The derivative of y is y'(x):
2 cosh(x y) sinh(x y) (y'(x) x+(d/dx(x)) y)+3 y^2 y'(x) = d/dx(3/2)
The derivative of x is 1:
3 y^2 y'(x)+2 cosh(x y) sinh(x y) (1 y+x y'(x)) = d/dx(3/2)
The derivative of 3/2 is zero:
3 y^2 y'(x)+2 cosh(x y) sinh(x y) (y+x y'(x)) = 0
Expand the left hand side:
2 cosh(x y) sinh(x y) y+2 x cosh(x y) sinh(x y) y'(x)+3 y^2 y'(x) = 0
Subtract 2 y sinh(x y) cosh(x y) from both sides:
2 x cosh(x y) sinh(x y) y'(x)+3 y^2 y'(x) = -2 cosh(x y) sinh(x y) y
Collect the left hand side in terms of y'(x):
(2 x cosh(x y) sinh(x y)+3 y^2) y'(x) = -2 cosh(x y) sinh(x y) y
Divide both sides by 2 x sinh(x y) cosh(x y)+3 y^2:
Answer: |y'(x) = -(2 cosh(x y) sinh(x y) y)/(2 x cosh(x y) sinh(x y)+3 y^2)
4: Sorry young person!. The computer software is unable to understand written questions!. If you can express it into something concrete to solve, then you might get somewhere.
