How do you use the quotient rule to differentiate #1 / (1 + x²)#?

Answer 1

The quotient rule states that the derivative of some function that's expressed as a quotient of two other functions, such as if #f(x)=(g(x))/(h(x))#, then the derivative of #f# is given through:

#f'(x)=(g'(x)h(x)-g(x)h'(x))/(h(x))^2#

For #f(x)=1/(1+x^2)#, we see that #g(x)=1# and #h(x)=1+x^2#.

We then see that #g'(x)=0# and #h'(x)=2x#. Plugging these in gives:

#f'(x)=(0(1+x^2)-1(2x))/(1+x^2)^2#

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

#" "#

Footnote

If you've learned the chain rule, it's easier to do this by rewriting the function as #(1+x^2)^-1# and then seeing that the derivative is #-(1+x^2)^-2d/dx(x^2)#, identical to what we found.

By signing up, you agree to our Terms of Service and Privacy Policy

Answer 2

Christopher Allers

To differentiate 1 / (1 + x²) using the quotient rule, apply the formula:

d/dx (u/v) = (v * du/dx - u * dv/dx) / v²

where u = 1 and v = (1 + x²).

Now, differentiate u and v with respect to x:

du/dx = 0 dv/dx = 2x

Now substitute these values into the quotient rule formula:

d/dx (1 / (1 + x²)) = ((1 + x²)(0) - 1(2x)) / (1 + x²)²

Simplify:

= (-2x) / (1 + x²)²

By signing up, you agree to our Terms of Service and Privacy Policy

Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.