Find the derivative of the following via implicit differentiation:
d/dx(y) = d/dx(-(x sqrt(36-x^2))+36 sin^(-1)(x/6))
The derivative of y is y'(x):
y'(x) = d/dx(-(x sqrt(36-x^2))+36 sin^(-1)(x/6))
Differentiate the sum term by term and factor out constants:
y'(x) = 36 d/dx(sin^(-1)(x/6))-d/dx(x sqrt(36-x^2))
Use the product rule, d/dx(u v) = v ( du)/( dx)+u ( dv)/( dx), where u = x and v = sqrt(36-x^2):
y'(x) = 36 (d/dx(sin^(-1)(x/6)))-sqrt(36-x^2) d/dx(x)+x d/dx(sqrt(36-x^2))
Simplify the expression:
y'(x) = -(sqrt(36-x^2) (d/dx(x)))-x (d/dx(sqrt(36-x^2)))+36 (d/dx(sin^(-1)(x/6)))
The derivative of x is 1:
y'(x) = -(x (d/dx(sqrt(36-x^2))))+36 (d/dx(sin^(-1)(x/6)))-1 sqrt(36-x^2)
Using the chain rule, d/dx(sqrt(36-x^2)) = ( dsqrt(u))/( du) ( du)/( dx), where u = 36-x^2 and ( d)/( du)(sqrt(u)) = 1/(2 sqrt(u)):
y'(x) = -sqrt(36-x^2)+36 (d/dx(sin^(-1)(x/6)))-(d/dx(36-x^2))/(2 sqrt(36-x^2)) x
Differentiate the sum term by term and factor out constants:
y'(x) = -sqrt(36-x^2)+36 (d/dx(sin^(-1)(x/6)))-(d/dx(36)-d/dx(x^2) x)/(2 sqrt(36-x^2))
The derivative of 36 is zero:
y'(x) = -sqrt(36-x^2)+36 (d/dx(sin^(-1)(x/6)))-(x (-(d/dx(x^2))+0))/(2 sqrt(36-x^2))
Simplify the expression:
y'(x) = -sqrt(36-x^2)+(x (d/dx(x^2)))/(2 sqrt(36-x^2))+36 (d/dx(sin^(-1)(x/6)))
Use the power rule, d/dx(x^n) = n x^(n-1), where n = 2: d/dx(x^2) = 2 x:
y'(x) = -sqrt(36-x^2)+36 (d/dx(sin^(-1)(x/6)))+(2 x x)/(2 sqrt(36-x^2))
Simplify the expression:
y'(x) = x^2/sqrt(36-x^2)-sqrt(36-x^2)+36 (d/dx(sin^(-1)(x/6)))
Using the chain rule, d/dx(sin^(-1)(x/6)) = ( dsin^(-1)(u))/( du) ( du)/( dx), where u = x/6 and ( d)/( du)(sin^(-1)(u)) = 1/sqrt(1-u^2):
y'(x) = x^2/sqrt(36-x^2)-sqrt(36-x^2)+36 (d/dx(x/6))/(sqrt(1-x^2/36))
Factor out constants:
y'(x) = x^2/sqrt(36-x^2)-sqrt(36-x^2)+(36 (d/dx(x))/6)/sqrt(1-x^2/36)
Simplify the expression:
y'(x) = x^2/sqrt(36-x^2)-sqrt(36-x^2)+(6 (d/dx(x)))/sqrt(1-x^2/36)
The derivative of x is 1:
y'(x) = x^2/sqrt(36-x^2)-sqrt(36-x^2)+(1 6)/sqrt(1-x^2/36)
Simplify the expression:
y'(x) = x^2/sqrt(36-x^2)-sqrt(36-x^2)+6/sqrt(1-x^2/36)
Expand the left hand side:
Answer: | y'(x) = x^2/sqrt(36-x^2) - sqrt(36-x^2) + 6/sqrt(1-x^2/36)
