Calculus one finding the derivative.

Question

I'm trying to work out the problem.

d/dx 2tan²x -sec²x the answer to the problem is = 2sec²xtanx. I don't know how to figure this one out if someone could please help me?

Answer 1

We know that 2tan^2(x) and sec^2(x) will be separate from each other as there is just subtraction between these two. With that in mind, let's first tackle with 2tan^2(x). We will need to use the chain rule. So first, that means we need to use the power rule, so we will get 4tan(x). Remember, then we leave the inner function the way it is, but then we also take the derivative of the inner function. Remember that the derivative of tan(x) is sec^2(x), so now we will have 4tan(x)sec^2(x)

Now sec^2(x), we will need to use the chain rule again, so we will use the power rule again, which will give us 2sex(x). Again, we will be leaving the inner function but will also take the derivative of the inner function. Remember that the derivative of sec(x) is sec(x)tan(x), so now we will have 2sex(x)sec(x)tan(x) = 2sec^2(x)tan(x).

So now with plugging these in, we have 4tan(x)sec^2(x) - 2sec^2(x)tan(x) = 2tan(x)sec^2(x) or 2sec^2(x)tan(x).

Answer 2

Take the derivatives of 2tan^2(x) and sec^2(x) separately.

First, we will find d/dx sec^2(x). Recall that d/dx (secx) = tanxsecx, so to find d/dx (sec^2(x)), we will use power rule, and then chain rule. Thus, we will bring out the 2 from the sec^2(x), reduce the power by 1 to get sec(x), so we get 2sec(x). Then apply chain rule, and multiply 2sec(x) by the derivative of sec(x), which is tan(x)sec(x). Thus, d/dx sec^2(x)=2sec^2(x)tan(x).

Next, we will find d/dx (2tan^2(x)). Recall that d/dx (tanx)=sec^2(x). Like before, will apply a power rule and chain rule. After the power rule, we would get 4tan(x), but then we multiply by the derivative of the inside function, the tan(x). Thus, d/dx (2tan^2(x))=4tan(x)sec^2(x).

Now we subtract the two: d/dx 2tan^2(x) - d/dx sec^2(x) = 4tan(x)sec^2(x)-2tan(x)sec^2(x)=2tan(x)sec^2(x).

Answer 3

(2tan^2(x) -sec^2(x))'

= 2(tan^2(x))' - (sec^2(x))'

= 4tanxsec^2(x) - -2secxsecxtanx

= 4tanxsec^2(x) - 2tanxsec^2(x)

=2tanxsec^2(x)