Find the derivative of f(x) = 2/(2t - 5)^2

Answer 1

Answer 2

In order to take the derivative of f(x) = 2 / (2t - 5)² , you have a couple of options. You can either:

1. Use the quotient rule, or
2. Move the binomial expression from the denominator to the numerator, and then apply the power rule and the chain rule.

In this example, I will show you how to use method #2.

Step 1: First, we'll need to rearrange our function so that we can get rid of the fraction. We can do this by using rules of exponents, which allows us to move the entire binomial from the denominator to the numerator, as long as we negate its exponent. Here is what that would look like:

f(x) = 2 (2t - 5)^-2 Now, we no longer have a fraction, but the exponent on (2t - 5) is now a -2.

Step 2: We are now ready to take the derivative. First, to apply the power rule, we will do three things:

1. Multiply the exponent on the binomial (the -2) by the coefficient of the binomial (which is 2), and write their product in front of the parentheses as the new coefficient.
2. Leave everything inside of the parentheses alone (for now), completely untouched.
3. Reduce the original exponent by subtracting 1 from it (so -2 -1).

This give us: -4 (2t - 5)^-3 **We're not done yet though!

Step 3: Now we will need to apply the chain rule, by taking the derivative of the binomial inside of the parentheses:

Derivative of the binomial in parentheses = 2

Step 4: Multiplying the derivative from Step 3 by the expression that we got from Step 2, we get our final answer:

f'(x) = -4 (2t - 5)^-3 (2) which can be simplified by multiplying the -4 and the 2:

f'(x) = -8 (2t - 5)^-3 Final answer