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Derivative of sin((pi/2+pin)x)

You have entered [src]

   //pi       \  \
sin||-- + pi*n|*x|
   \\2        /  /

$$\sin{\left(x \left(\pi n + \frac{\pi}{2}\right) \right)}$$

sin((pi/2 + pi*n)*x)

Detail solution

Let $u = x \left(\pi n + \frac{\pi}{2}\right)$ .
The derivative of sine is cosine:

$\frac{d}{d u} \sin{\left(u \right)} = \cos{\left(u \right)}$
Then, apply the chain rule. Multiply by $\frac{\partial}{\partial x} x \left(\pi n + \frac{\pi}{2}\right)$ :
1. The derivative of a constant times a function is the constant times the derivative of the function.
  1. Apply the power rule: $x$ goes to $1$
  So, the result is: $\pi n + \frac{\pi}{2}$
The result of the chain rule is:

$\left(\pi n + \frac{\pi}{2}\right) \cos{\left(x \left(\pi n + \frac{\pi}{2}\right) \right)}$
Now simplify:

$\frac{\pi \left(2 n + 1\right) \cos{\left(\pi x \left(n + \frac{1}{2}\right) \right)}}{2}$

The answer is:

$\frac{\pi \left(2 n + 1\right) \cos{\left(\pi x \left(n + \frac{1}{2}\right) \right)}}{2}$

The first derivative [src]

/pi       \    //pi       \  \
|-- + pi*n|*cos||-- + pi*n|*x|
\2        /    \\2        /  /

$$\left(\pi n + \frac{\pi}{2}\right) \cos{\left(x \left(\pi n + \frac{\pi}{2}\right) \right)}$$

Simplify

The second derivative [src]

   2          2                    
-pi *(1/2 + n) *sin(pi*x*(1/2 + n))

$$- \pi^{2} \left(n + \frac{1}{2}\right)^{2} \sin{\left(\pi x \left(n + \frac{1}{2}\right) \right)}$$

Simplify

The third derivative [src]

   3          3                    
-pi *(1/2 + n) *cos(pi*x*(1/2 + n))

$$- \pi^{3} \left(n + \frac{1}{2}\right)^{3} \cos{\left(\pi x \left(n + \frac{1}{2}\right) \right)}$$

Simplify

Derivative of sin((pi/2+pi*n)*x)