How do you find the derivative of $sin^2(8x)-(pi)x$ ?

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Answer 1 · 2017-08-13T22:27:55

$frac(d)(dx) ((sin(8 x))^(2) - pi x) = 8 sin(16 x) - pi$

We have: $sin^(2)(8 x) - (pi) x$

$= (sin(8 x))^(2) - pi x$

First, let's apply the difference rule:

$Rightarrow frac(d)(dx) ((sin(8 x))^(2) - pi x) = frac(d)(dx)((sin(8 x))^(2)) - frac(d)(dx)(pi x)$

$Rightarrow frac(d)(dx) ((sin(8 x))^(2) - pi x) = frac(d)(dx) ((sin(8 x))^(2)) - pi$

Then, let's use the chain rule.

Let $u = sin(8 x) Rightarrow u' = cos(8 x)$ , $v = u^(2) Rightarrow v' = 2 u$ and $w = 8 x Rightarrow w' = 8$ :

$Rightarrow frac(d)(dx) ((sin(8 x))^(2) - pi x) = u' cdot v' cdot w' - pi$

$Rightarrow frac(d)(dx) ((sin(8 x))^(2) - pi x) = cos(8 x) cdot 2 u cdot 8 - pi$

$Rightarrow frac(d)(dx) ((sin(8 x))^(2) - pi x) = 8 cdot 2 u cos(8 x) - pi$

Now, let's replace $u$ with $sin(8 x)$ :

$Rightarrow frac(d)(dx) ((sin(8 x))^(2) - pi x) = 8 cdot 2 sin(8 x) cos(8 x) - pi$

Let's apply the double angle formula for $sin(x)$ ; $sin(2 x) = 2 sin(x) cos(x)$ :

$Rightarrow frac(d)(dx) ((sin(8 x))^(2) - pi x) = 8 cdot sin(2 cdot 8 x) - pi$

$therefore frac(d)(dx) ((sin(8 x))^(2) - pi x) = 8 sin(16 x) - pi$

How do you find the derivative of sin^2(8x)-(pi)xsin^2(8x)-(pi)x?