How do you differentiate #f(x)=1/sqrt(1-x)# using the chain rule?

Question

Emily Ammerman · Accepted Answer

#f'(x)=1/(2sqrt((1-x)^3))#

Rewrite using negative exponents:

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

Then, applying the Power Rule in conjunction with the Chain Rule,

#f'(x)=-1/2(1-x)^(-3/2)*d/dx(1-x)#

#d/dx(1-x)=-1,# so we have

#f'(x)=1/(2(1-x)^(3/2))#

Rewriting in radical form:

#f'(x)=1/(2sqrt((1-x)^3))#

Axel Alvarez · Answer

undefined To differentiate $ f(x) = \frac{1}{\sqrt{1-x}} $ using the chain rule, follow these steps:

1. Identify the outer function and the inner function. In this case, the outer function is $ \frac{1}{\sqrt{x}} $ and the inner function is $ 1-x $.

2. Differentiate the outer function with respect to its variable, treating the inner function as if it were a simple variable. The derivative of $ \frac{1}{\sqrt{x}} $ is $ -\frac{1}{2}x^{-3/2} $.

3. Multiply the result from step 2 by the derivative of the inner function with respect to its variable. The derivative of $ 1-x $ is $ -1 $.

4. Combine the results to get the derivative of the entire function. So, the derivative of $ f(x) = \frac{1}{\sqrt{1-x}} $ is $ -\frac{1}{2}(1-x)^{-3/2} $.