The function we are examining is 2 sin(x) cos(x). To find the first and second derivatives, we will apply differentiation rules.
Step 1: First Derivative
We can use the product rule for derivatives, which states that if you have two functions u and v, the derivative of their product is given by:
(u * v)’ = u’v + uv’
Let u = 2 sin(x) and v = cos(x). Then we find:
- u’ = 2 cos(x) (derivative of 2 sin(x))
- v’ = -sin(x) (derivative of cos(x))
Applying the product rule:
f'(x) = u'v + uv'
= (2 cos(x))cos(x) + (2 sin(x))(-sin(x))
= 2 cos²(x) - 2 sin²(x)Thus, the first derivative is:
f'(x) = 2 cos²(x) - 2 sin²(x)Step 2: Second Derivative
Now, we need to differentiate f'(x) = 2 cos²(x) – 2 sin²(x). We apply the chain rule and product rule again:
For each term, we can differentiate:
- For 2 cos²(x), we get:
d/dx [2 cos²(x)] = 2 * 2 cos(x)(-sin(x)) = -4 cos(x) sin(x)d/dx [-2 sin²(x)] = -2 * 2 sin(x)(cos(x)) = -4 sin(x) cos(x)Combining these results:
f''(x) = -4 cos(x) sin(x) - 4 sin(x) cos(x) = -8 sin(x) cos(x)The second derivative is:
f''(x) = -8 sin(x) cos(x)Summary:
- First Derivative:
f'(x) = 2 cos²(x) - 2 sin²(x) - Second Derivative:
f''(x) = -8 sin(x) cos(x)