derivative of y=2x(100-x^2)^1/2?

Question

Please help me, I have trouble simplifying it.

Answer 1

d(2x√(100-x²))/dx

Use the product rule:

d(2x)/dx √(100-x²) + 2x d(√(100-x²))/dx

Use the power rule and chain rule:

2√(100-x²) + 2x*1/2 * (100-x²)^(-1/2) * d(100-x²)/dx

Simplify:

2√(100-x²) + x/√(100-x²) * d(100-x²)/dx

Use the power rule:

2√(100-x²) + x/√(100-x²) * (-2x)

Simplify:

2√(100-x²) - 2x²/√(100-x²)

And we are done.

Answer 2

Ok, we might use the product rule.

y= uv

y' = u'v + v'u

where u and v are the parts of the product in the function.

In this function, we can have
u = 2x
u' = 2

v=(100-x^2)^(1/2)
v' = 1/2(100-x^2)^(-1/2) * -2x
= -2x*1/2(100-x^2)^(-1/2) (using the function of a function rule)
= -x (100-x^2)^(-1/2)

When we apply the rule: y'= u'v +v'u

= 2(100-x^2)^(1/2) + -x (100-x^2)^(-1/2) * 2x

= 2(100-x^2)^(1/2) - 2x^2 (100-x^2)^(-1/2)

This bit is probably better done on paper... but hopefully this gets the idea across...

To get a common denominator of root(100-x^2), we have to multiply the first term by root(100-x^2).

= [2(root)(100-x^2)* (root)(100-x^2) - 2x^2]/(root)(100 -x^2)

= [2(100-x^2) - 2x^2] / (root)(100 -x^2)

= (200 -4x^2) /(root)(100 -x^2)

That should be a right answer. Hope it helps. Derivatives are tricky... don't let it bother you.

Answer 3

x(100-x^2)^-1/2 times (-2x)

equals -2x^2(100-x^2)^-1/2

Answer 4

dy/dx = 2(100-x^2)^(1/2) + 2x (1/2)(100-x^2)^(-1/2) (-2x)
write (100-x^2)^(1/2) as (100-x^2)^(-1/2) . (100-x^2) so we can take out a common factor of (100-x^2)^(-1/2):
= (100-x^2)^(-1/2) [(2(100-x^2) + 2x (1/2) (-2x)]
= (100-x^2)^(-1/2) . (200 - 2x^2 - 2x^2)
= (200 - 4x^2) / √(100-x^2).

Answer 5

y=2x(100-x^2)^1/2

y' = 2x [1/2 (100 - x^2)^(-1/2) (-2x)] + 2 (100 - x^2)^1/2

y' = -2x^2 (100 - x^2)^(-1/2) + 2 (100 - x^2)^1/2

y' = 2 (100-x^2)^(-1/2) [(-x^2) + (100 - x^2)]

y' = 2 (100 - 2x^2)/(100-x^2)^(1/2)

Answer 6

-4(x^2-50)/(100-x^2)^1/2

Answer 7

y=2x(100-x^2)^1/2 = (400x^2 - 4x^4)^1/2

y' = 1/2(400x^2 - 4x^4)^(-1/2)*(800x - 16x^3) =

= 16x(50 - x^2)/(4x*(100 - x^2)^1/2) =

= 4(50 - x^2)/(100 - x^2)^(1/2)

Answer 8

f '(x) = 5 more often than not the spinoff of ax^n is nax^(n-a million). right here a = 5 and n = a million. The spinoff of a consistent (-8 right here) is 0. imagine about the graph. y = 5x - 8 is the equation of a immediately line and the gradient is 5.