y = -14sin(8x)2cos(5x) - 4sin(5x)cos(8x)
u = -14sin(8x)
u' = -112cos(8x)
v = 2cos(5x)
v' = -10sin(5x)
u2 = -4sin(5x)
u'2 = -20cos(5x)
v2 = cos(8x)
v'2 = -8sin(8x)
y' = (uv') + (vu') + (u2v'2) + (v2u'2)
y' = [(-14sin(8x))(-10sin(5x)] - [(112cos(8x)(2cos(5x)] - [4sin(5x))(-8sin(8x)] - [20cos(5x))(cos(8x)]
Is all of this correct? If not, please tell me where I have made a mistake.
Thank you.
Yes, but you still need to simplify the terms.
Yes, your calculations are correct up to this point. You have correctly found the derivatives of the individual terms and used them to calculate the derivative of the function y. The expression for y' that you have written is also correct.
To simplify further, you can distribute and combine like terms within each bracket. Let's go through the simplification process step by step:
y' = [(-14sin(8x))(-10sin(5x)] - [(112cos(8x)(2cos(5x)] - [4sin(5x))(-8sin(8x)] - [20cos(5x))(cos(8x)]
Expanding each bracket:
y' = [140sin(8x)sin(5x)] - [224cos(8x)cos(5x)] - [32sin(5x)sin(8x)] - [20cos(5x)cos(8x)]
Now, we can use trigonometric identities to simplify further:
1. sin(a)sin(b) = (1/2)[cos(a-b) - cos(a+b)]
2. cos(a)cos(b) = (1/2)[cos(a-b) + cos(a+b)]
Applying these identities:
y' = (1/2)[70(cos(8x-5x) - cos(8x+5x))] - (1/2)[112(cos(8x+5x) + cos(8x-5x))] - (1/2)[16(cos(8x-5x) - cos(8x+5x))] - (1/2)[10(cos(8x+5x) + cos(8x-5x))]
Simplifying the terms:
y' = (1/2)[70cos(3x) - 112cos(13x) - 16cos(3x) - 10cos(13x)]
Combining like terms:
y' = (1/2)[-114cos(3x) - 122cos(13x)]
So, the simplified expression for y' is:
y' = -57cos(3x) - 61cos(13x)
Therefore, your final answer for y' should be:
y' = -57cos(3x) - 61cos(13x)