If f(x)=sin^2(3-x) then f'(0)=
A. -2cos3
B. -2sin3cos3
C. 6cos3
D. 2sin3cos3
E. 6sin3cos3
I got B
Yes, B is correct.
f'(x)
d(sin^2(3-x))/dx
Apply chain rule
=2sin(3-x) dsin(3-x)/dx
=2sin(3-x)cos(3-x)d(3-x)/dx
=-2sin(3-x)cos(3-x)
Then calculate f'(0) from above.
Well, I must say, you're really sprinkling some mathematical spice onto my clownish demeanor! So, let's juggle with some derivatives, shall we?
To find the derivative of f(x)=sin^2(3-x), we'll use the chain rule. The chain rule states that the derivative of f(g(x)) equals f'(g(x)) * g'(x).
First, let's find the derivative of sin^2(3-x) with respect to (3-x). That would be 2*sin(3-x) * cos(3-x), thanks to the power rule and the chain rule.
Now, we have to find the derivative of (3-x) with respect to x. That's simply -1, since x is decreasing by 1.
Finally, we multiply these two derivatives together: 2*sin(3-x) * cos(3-x) * -1.
Okay, now that we've established that, let's substitute x=0 into our derivative to find f'(0):
f'(0) = 2*sin(3-0) * cos(3-0) * -1
= 2*sin(3) * cos(3) * -1
= -2*sin(3)*cos(3)
And there you have it, my friend! The answer is not B, but A! So, A is the real "culprit" here. Keep up the mathematical juggling, my clever friend!
To find the derivative of the function f(x) = sin^2(3-x), we can use the chain rule.
First, we need to take the derivative of the outer function (sin^2(3-x)), keeping the inner function (3-x) unchanged.
The derivative of sin(u) is cos(u), so using the chain rule, the derivative of sin^2(3-x) is 2sin(3-x)cos(3-x).
Now, to find f'(0), we substitute x=0 into the derivative we just found.
f'(0) = 2sin(3-0)cos(3-0) = 2sin(3)cos(3) = 2sin3cos3.
Therefore, the correct answer is D. 2sin3cos3.
To find the derivative of the function f(x) = sin^2(3 - x) and then evaluate it at x = 0, you can use the chain rule.
Step 1: Find the derivative of the outer function.
The derivative of sin^2(u) with respect to u can be found using the chain rule. Let u = (3 - x), so the derivative of sin^2(u) with respect to u is 2sin(u) * cos(u).
Step 2: Find the derivative of the inner function.
The derivative of (3 - x) with respect to x is -1.
Step 3: Apply the chain rule.
Multiply the derivative of the outer function (2sin(u) * cos(u)) by the derivative of the inner function (-1).
f'(x) = -2sin(3 - x) * cos(3 - x)
Step 4: Evaluate f'(0).
Plug in x = 0 into the derivative expression to find f'(0).
f'(0) = -2sin(3 - 0) * cos(3 - 0)
= -2sin(3) * cos(3)
= -2sin3 * cos3
Therefore, the correct answer is B. -2sin3cos3.