derivative:

4/(1-2x^2)

Use the general formula for deriving fractions.

I remember a nmemonic like "lo di hi less hi di lo all over lo squared".

...or write it as 4*(1-2x^2)^(-1) and use the chain rule.

To find the derivative of the function 4/(1-2x^2), we can use the general formula for differentiating fractions. The formula is often remembered using a mnemonic: "lo di hi less hi di lo all over lo squared." Let's break it down step by step:

1. Start by identifying the different parts of the fraction:
- The "lo" stands for the denominator (1 - 2x^2).
- The "di" stands for differentiating the numerator, which is 4.
- The "hi" stands for leaving the denominator as it is.
- The "less" signifies subtracting the second part.
- The "hi di lo" stands for differentiating the denominator.
- Finally, "all over lo squared" means dividing everything by the square of the denominator.

Using this mnemonic, we can write the steps for differentiating the given function:

1. Identify the different parts:
lo: 1 - 2x^2 (the denominator)
di: 4 (the numerator)
hi: 1 - 2x^2 (the denominator)

2. Apply the formula:
(lo di hi - hi di lo) / lo^2

3. Substitute in the values:
[(1 - 2x^2)(4) - (1 - 2x^2)(0)] / (1 - 2x^2)^2

The second part, "hi di lo," equals 0 because differentiating a constant (1) gives us 0.

4. Simplify the expression:
(4 - 8x^2) / (1 - 2x^2)^2

Therefore, the derivative of 4/(1 - 2x^2) is (4 - 8x^2) / (1 - 2x^2)^2.

Alternatively, you can rewrite the original function as 4*(1 - 2x^2)^(-1) and use the chain rule to differentiate it. However, the result will be the same as the previous method:

1. Rewrite the function as 4*(1 - 2x^2)^(-1).

2. Use the chain rule:
Take the derivative of the function inside the parentheses, which is -4x.
Multiply it by the derivative of the exponent, which is (-1)*(1 - 2x^2)^(-2)*(-4x).

3. Simplify the expression by canceling out common factors:
(-4x) / (1 - 2x^2)^2

Again, the result is (4 - 8x^2) / (1 - 2x^2)^2, which matches the previous method.