y=√(1+x÷√(1-x) differentiate the function w.r.t.'x'
Answers
Answer:
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Step-by-step explanation:
Find the derivative of the following via implicit differentiation:
d/dx(y) = d/dx(sqrt(1 + x/sqrt(1 - x)))
Using the chain rule, d/dx(y) = ( dy(u))/( du) ( du)/( dx), where u = x and d/( du)(y(u)) = y'(u):
d/dx(x) y'(x) = d/dx(sqrt(1 + x/sqrt(1 - x)))
The derivative of x is 1:
1 y'(x) = d/dx(sqrt(1 + x/sqrt(1 - x)))
Using the chain rule, d/dx(sqrt(x/sqrt(1 - x) + 1)) = ( dsqrt(u))/( du) ( du)/( dx), where u = x/sqrt(1 - x) + 1 and d/( du)(sqrt(u)) = 1/(2 sqrt(u)):
y'(x) = (d/dx(1 + (1 - x)^(-1/2) x))/(2 sqrt(1/sqrt(1 - x) x + 1))
Differentiate the sum term by term:
y'(x) = d/dx(1) + d/dx((1 - x)^(-1/2) x) 1/(2 sqrt(1 + x/sqrt(1 - x)))
The derivative of 1 is zero:
y'(x) = (d/dx(x/sqrt(1 - x)) + 0)/(2 sqrt(1 + x/sqrt(1 - x)))
Simplify the expression:
y'(x) = (d/dx((1 - x)^(-1/2) x))/(2 sqrt(1 + x/sqrt(1 - x)))
Use the product rule, d/dx(u v) = v ( du)/( dx) + u ( dv)/( dx), where u = 1/sqrt(1 - x) and v = x:
y'(x) = x d/dx((1 - x)^(-1/2)) + (d/dx(x))/(sqrt(1 - x)) 1/(2 sqrt(1 + x/sqrt(1 - x)))
Using the chain rule, d/dx(1/sqrt(1 - x)) = d/( du)1/sqrt(u) ( du)/( dx), where u = 1 - x and d/( du)(1/sqrt(u)) = -1/(2 u^(3/2)):
y'(x) = ((d/dx(x))/sqrt(1 - x) + -(d/dx(1 - x))/(2 (1 - x)^(3/2)) x)/(2 sqrt(1 + x/sqrt(1 - x)))
Differentiate the sum term by term and factor out constants:
y'(x) = ((d/dx(x))/sqrt(1 - x) - d/dx(1) - d/dx(x) x/(2 (1 - x)^(3/2)))/(2 sqrt(1 + x/sqrt(1 - x)))
The derivative of 1 is zero:
y'(x) = ((d/dx(x))/sqrt(1 - x) - (x (-(d/dx(x)) + 0))/(2 (1 - x)^(3/2)))/(2 sqrt(1 + x/sqrt(1 - x)))
Simplify the expression:
y'(x) = ((d/dx(x))/sqrt(1 - x) + (x (d/dx(x)))/(2 (1 - x)^(3/2)))/(2 sqrt(1 + x/sqrt(1 - x)))
The derivative of x is 1:
y'(x) = ((d/dx(x))/sqrt(1 - x) + 1 x/(2 (1 - x)^(3/2)))/(2 sqrt(1 + x/sqrt(1 - x)))
The derivative of x is 1:
y'(x) = (x/(2 (1 - x)^(3/2)) + 1 1/sqrt(1 - x))/(2 sqrt(1 + x/sqrt(1 - x)))
Factor the numerator and denominator of the right hand side:
y'(x) = (-2 + x)/((2 sqrt(1 - x) (-1 + x)) (2 sqrt(1 + x/sqrt(1 - x))))
Cancel common terms in the numerator and denominator:
Answer: |
| y'(x) = (-2 + x)/(4 sqrt(1 - x) (-1 + x) sqrt(1 + x/sqrt(1 - x)))