Question

Determine the first and second derivative of sin^2t-2cos^2t.

Answer 1

My answer : First derivative: 2sintcost+4costsint
second derivative: -2costsint+4sintcost

Answer 2

But it's wrong.

Answer 3

@sleepyhead314

Answer 4

do you know why it's wrong? It's not wrong, just not the simplest form

Answer 5

add them together, you have 6sintcost, right? for the first derivative

Answer 6

and it is = 3(2 sint cos t) right?

Answer 7

= 3 sin(2t) . that is the simplest form

Answer 8

then, take the second derivative

Answer 9

Second derivative is incorrect. Use the product rule.

Alternatively, you may wish to use identities for the given function to condense the work:
\[\sin^2t-2\cos^2t=\frac{1-\cos2t}{2}-2\left(\frac{1+\cos2t}{2}\right)=-\frac{1}{2}-\frac{3}{2}\cos2t\]
Then the first derivative is
\[3\sin2t=3(2\sin t\cos t)=6\sin t\cos t\]
and the second derivative is
\[6\cos2t=6\cos^2t-6\sin^2t=\text{whatever else this may be equivalent to}\]