Calculate the derivative with respect to x x5y + x3 y 5x + 4y
Find the derivative of the following via implicit differentiation: d/dx(5 x^3 y+x^5 y) = d/dx(5 x+4 y) Differentiate the sum term by term and factor out constants: 5 d/dx(x^3 y)+d/dx(x^5 y) = d/dx(5 x+4 y) Use the product rule, d/dx(u v) = v ( du)/( dx)+u ( dv)/( dx), where u = x^3 and v = y: d/dx(x^5 y)+5 x^3 d/dx(y)+d/dx(x^3) y = d/dx(5 x+4 y) The derivative of y is y\'(x): d/dx(x^5 y)+5 (y\'(x) x^3+(d/dx(x^3)) y) = d/dx(5 x+4 y) Use the product rule, d/dx(u v) = v ( du)/( dx)+u ( dv)/( dx), where u = x^5 and v = y: x^5 d/dx(y)+d/dx(x^5) y+5 ((d/dx(x^3)) y+x^3 y\'(x)) = d/dx(5 x+4 y) Simplify the expression: x^5 (d/dx(y))+(d/dx(x^5)) y+5 ((d/dx(x^3)) y+x^3 y\'(x)) = d/dx(5 x+4 y) The derivative of y is y\'(x): y\'(x) x^5+(d/dx(x^5)) y+5 ((d/dx(x^3)) y+x^3 y\'(x)) = d/dx(5 x+4 y) Use the power rule, d/dx(x^n) = n x^(n-1), where n = 5: d/dx(x^5) = 5 x^4: 5 x^4 y+x^5 y\'(x)+5 ((d/dx(x^3)) y+x^3 y\'(x)) = d/dx(5 x+4 y) Use the power rule, d/dx(x^n) = n x^(n-1), where n = 3: d/dx(x^3) = 3 x^2: 5 x^4 y+x^5 y\'(x)+5 (3 x^2 y+x^3 y\'(x)) = d/dx(5 x+4 y) Differentiate the sum term by term and factor out constants: 5 x^4 y+x^5 y\'(x)+5 (3 x^2 y+x^3 y\'(x)) = 5 d/dx(x)+4 d/dx(y) The derivative of x is 1: 5 x^4 y+x^5 y\'(x)+5 (3 x^2 y+x^3 y\'(x)) = 4 (d/dx(y))+1 5 The derivative of y is y\'(x): 5 x^4 y+x^5 y\'(x)+5 (3 x^2 y+x^3 y\'(x)) = 5+4 y\'(x) Expand the left hand side: 15 x^2 y+5 x^4 y+5 x^3 y\'(x)+x^5 y\'(x) = 5+4 y\'(x) Subtract 4 y\'(x) from both sides: 15 x^2 y+5 x^4 y-4 y\'(x)+5 x^3 y\'(x)+x^5 y\'(x) = 5 Subtract 5 x^4 y+15 x^2 y from both sides: -4 y\'(x)+5 x^3 y\'(x)+x^5 y\'(x) = 5-15 x^2 y-5 x^4 y Collect the left hand side in terms of y\'(x): (-4+5 x^3+x^5) y\'(x) = 5-15 x^2 y-5 x^4 y Divide both sides by x^5+5 x^3-4: Answer: | | y\'(x) = (5-15 x^2 y-5 x^4 y)/(-4+5 x^3+x^5)
Homework Sourse