Solve sin(mu_{0})

$$\sin ( \mu _ { 0 } )$$

Differentiate w.r.t. μ_0

$$\cos(\mu _{0})$$

Steps Using Definition of a Derivative

For a function f\left(x\right), the derivative is the limit of \frac{f\left(x+h\right)-f\left(x\right)}{h} as h goes to 0, if that limit exists.

$$\frac{\mathrm{d}}{\mathrm{d}\mu _{0}}(\sin(\mu _{0}))=\left(\lim_{h\to 0}\frac{\sin(\mu _{0}+h)-\sin(\mu _{0})}{h}\right)$$

Use the Sum Formula for Sine.

$$\lim_{h\to 0}\frac{\sin(h+\mu _{0})-\sin(\mu _{0})}{h}$$

Factor out \sin(\mu _{0}).

$$\lim_{h\to 0}\frac{\sin(\mu _{0})\left(\cos(h)-1\right)+\cos(\mu _{0})\sin(h)}{h}$$

Rewrite the limit.

$$\left(\lim_{h\to 0}\sin(\mu _{0})\right)\left(\lim_{h\to 0}\frac{\cos(h)-1}{h}\right)+\left(\lim_{h\to 0}\cos(\mu _{0})\right)\left(\lim_{h\to 0}\frac{\sin(h)}{h}\right)$$

Use the fact that \mu _{0} is a constant when computing limits as h goes to 0.

$$\sin(\mu _{0})\left(\lim_{h\to 0}\frac{\cos(h)-1}{h}\right)+\cos(\mu _{0})\left(\lim_{h\to 0}\frac{\sin(h)}{h}\right)$$

The limit \lim_{\mu _{0}\to 0}\frac{\sin(\mu _{0})}{\mu _{0}} is 1.

$$\sin(\mu _{0})\left(\lim_{h\to 0}\frac{\cos(h)-1}{h}\right)+\cos(\mu _{0})$$

To evaluate the limit \lim_{h\to 0}\frac{\cos(h)-1}{h}, first multiply the numerator and denominator by \cos(h)+1.

$$\left(\lim_{h\to 0}\frac{\cos(h)-1}{h}\right)=\left(\lim_{h\to 0}\frac{\left(\cos(h)-1\right)\left(\cos(h)+1\right)}{h\left(\cos(h)+1\right)}\right)$$

Multiply \cos(h)+1 times \cos(h)-1.

$$\lim_{h\to 0}\frac{\left(\cos(h)\right)^{2}-1}{h\left(\cos(h)+1\right)}$$

Use the Pythagorean Identity.

$$\lim_{h\to 0}-\frac{\left(\sin(h)\right)^{2}}{h\left(\cos(h)+1\right)}$$

Rewrite the limit.

$$\left(\lim_{h\to 0}-\frac{\sin(h)}{h}\right)\left(\lim_{h\to 0}\frac{\sin(h)}{\cos(h)+1}\right)$$

The limit \lim_{\mu _{0}\to 0}\frac{\sin(\mu _{0})}{\mu _{0}} is 1.

$$-\left(\lim_{h\to 0}\frac{\sin(h)}{\cos(h)+1}\right)$$

Use the fact that \frac{\sin(h)}{\cos(h)+1} is continuous at 0.

$$\left(\lim_{h\to 0}\frac{\sin(h)}{\cos(h)+1}\right)=0$$

Substitute the value 0 into the expression \sin(\mu _{0})\left(\lim_{h\to 0}\frac{\cos(h)-1}{h}\right)+\cos(\mu _{0}).

$$\cos(\mu _{0})$$

Evaluate

$$\sin(\mu _{0})$$