How do you use the Product Rule to find the derivative of # y = xsqrt(1-x^2)#?

Question

Isaiah Allen · Accepted Answer

#color(red)(dy/dx= sqrt(1-x^2)-x^2/sqrt(1-x^2))#

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

The Product Rule, states that if #y=uv#, then

#dy/dx = u(dv)/dx+v(du)/dx#

Let #u=x# and #v=(1-x^2)^(1/2)#

Then

#(du)/dx=1#

#(dv)/dx=1/2(1-x^2)^(-1/2)(-2x)= -x(1-x^2)^(-1/2)#

#dy/dx = u(dv)/dx+v(du)/dx#

#dy/dx=-x^2(1-x^2)^(-1/2) +(1-x^2)^(1/2)#

#dy/dx= sqrt(1-x^2)-x^2/sqrt(1-x^2)#

Emilia Aguilar · Answer

undefined To use the Product Rule to find the derivative of $ y = x\sqrt{1-x^2} $, you can follow these steps:

1. Identify the functions being multiplied together: $ f(x) = x $ and $ g(x) = \sqrt{1-x^2} $.
2. Apply the Product Rule formula, which states: $ (f \cdot g)' = f' \cdot g + f \cdot g' $.
3. Find the derivatives of $ f(x) $ and $ g(x) $, denoted as $ f'(x) $ and $ g'(x) $, respectively.
4. Substitute the functions and their derivatives into the Product Rule formula.
5. Simplify the expression to obtain the derivative of $ y $.

Derivative of $ f(x) = x $: $ f'(x) = 1 $.
Derivative of $ g(x) = \sqrt{1-x^2} $: $ g'(x) = \frac{-x}{\sqrt{1-x^2}} $.

Now apply the Product Rule:

$ y' = f' \cdot g + f \cdot g' $
$ y' = 1 \cdot \sqrt{1-x^2} + x \cdot \frac{-x}{\sqrt{1-x^2}} $

Simplify:

$ y' = \sqrt{1-x^2} - \frac{x^2}{\sqrt{1-x^2}} $