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Differentiate w.r.t. h
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\frac{\mathrm{d}}{\mathrm{d}h}(\sin(h))=\left(\lim_{t\to 0}\frac{\sin(h+t)-\sin(h)}{t}\right)
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.
\lim_{t\to 0}\frac{\sin(t+h)-\sin(h)}{t}
Use the Sum Formula for Sine.
\lim_{t\to 0}\frac{\sin(h)\left(\cos(t)-1\right)+\cos(h)\sin(t)}{t}
Factor out \sin(h).
\left(\lim_{t\to 0}\sin(h)\right)\left(\lim_{t\to 0}\frac{\cos(t)-1}{t}\right)+\left(\lim_{t\to 0}\cos(h)\right)\left(\lim_{t\to 0}\frac{\sin(t)}{t}\right)
Rewrite the limit.
\sin(h)\left(\lim_{t\to 0}\frac{\cos(t)-1}{t}\right)+\cos(h)\left(\lim_{t\to 0}\frac{\sin(t)}{t}\right)
Use the fact that h is a constant when computing limits as t goes to 0.
\sin(h)\left(\lim_{t\to 0}\frac{\cos(t)-1}{t}\right)+\cos(h)
The limit \lim_{h\to 0}\frac{\sin(h)}{h} is 1.
\left(\lim_{t\to 0}\frac{\cos(t)-1}{t}\right)=\left(\lim_{t\to 0}\frac{\left(\cos(t)-1\right)\left(\cos(t)+1\right)}{t\left(\cos(t)+1\right)}\right)
To evaluate the limit \lim_{t\to 0}\frac{\cos(t)-1}{t}, first multiply the numerator and denominator by \cos(t)+1.
\lim_{t\to 0}\frac{\left(\cos(t)\right)^{2}-1}{t\left(\cos(t)+1\right)}
Multiply \cos(t)+1 times \cos(t)-1.
\lim_{t\to 0}-\frac{\left(\sin(t)\right)^{2}}{t\left(\cos(t)+1\right)}
Use the Pythagorean Identity.
\left(\lim_{t\to 0}-\frac{\sin(t)}{t}\right)\left(\lim_{t\to 0}\frac{\sin(t)}{\cos(t)+1}\right)
Rewrite the limit.
-\left(\lim_{t\to 0}\frac{\sin(t)}{\cos(t)+1}\right)
The limit \lim_{h\to 0}\frac{\sin(h)}{h} is 1.
\left(\lim_{t\to 0}\frac{\sin(t)}{\cos(t)+1}\right)=0
Use the fact that \frac{\sin(t)}{\cos(t)+1} is continuous at 0.
\cos(h)
Substitute the value 0 into the expression \sin(h)\left(\lim_{t\to 0}\frac{\cos(t)-1}{t}\right)+\cos(h).