So I am suppose to evaulate this problem y=tan^4(2x) and I am confused.

my friend did this : 3 tan ^4 (2x) d sec^ 2x (2x)= 6 tan ^4 (2x) d sec^2 (2x)
She says it's right but what confuses me is she deriving the 4 and made it a three? I did the problem like this:
tan^4 (2x)= 4 tan^3 (2x) d sec ^2 (2x)(2x)= 8 tan^3 (2x) d sec^2 (2x)

Can anyone explain this to me?

Go on :

wolframalpha dot com

Whe page be open in rectangle type :

derivative tan^4(2x)

then click option =

After few seconds when you see result click option :

Show steps

To evaluate the expression y = tan^4(2x), both you and your friend attempted to calculate the derivative of the expression. However, there seems to be confusion regarding the exponents.

Let's break down each step:

Your friend's approach:
1. Starting with y = tan^4(2x), she applied the power rule for derivatives to the tangent function, which states that d/dx(tan(x)) = sec^2(x). In this case, she applied the power rule four times, resulting in 4 tan^3(2x) d/dx(tan(2x)).
2. Then, she multiplied by d/dx(sec^2(2x)), which correctly takes into account the chain rule.
3. As a result, your friend obtained 3 tan^4(2x) d/dx(sec^2(2x)).

Your approach:
1. You correctly applied the power rule three times, resulting in 4 tan^3(2x) d/dx(tan(2x)).
2. Similar to your friend, you then multiplied by d/dx(sec^2(2x)).
3. Consequently, you obtained 8 tan^3(2x) d/dx(sec^2(2x)).

The discrepancy between your answers arises from a difference in the application of the power rule. In this case, it seems that your approach is the correct one.

To clarify, the power rule for derivative states that if y = a^n, where 'a' is a constant and 'n' is a real number, then dy/dx = n * a^(n-1) * (derivative of 'a' with respect to 'x').

In your case, a = tan(2x) and n = 4. Therefore, when taking the derivative, you should apply the power rule as follows:

dy/dx = 4 * tan^3(2x) * (derivative of tan(2x) with respect to 'x').

By following this approach, you arrived at the correct answer of 8 tan^3(2x) d/dx(sec^2(2x)).

Remember to pay attention to the correct application of mathematical rules and formulas when solving derivative problems.