Question

find the derivative of sin(ax) using the limit definition and trig sum identities.

Answer 1

\[\lim_{h \rightarrow 0} [\sin(ax)\cos(ah)+\sin(ah)\cos(ax)-\sin(ax)]/h\]

Answer 2

where do I go from here?

Answer 3

Use a different trig identity. The idea is to get rid of the subtraction and addition signs and be left with only multiplications/divisions. Then we use the limit law for multiplication (the limit of a product is the product of the limits). See below:\[=\lim_{h \rightarrow 0}\frac{2\cos (\frac{2ax+ah}{2})\sin (\frac{ah}{2})}{h}\]we get:\[=2\lim_{h \rightarrow 0}\cos (ax+\frac{ah}{2})*\lim_{h \rightarrow 0}\frac{\sin (\frac{ah}{2})}{h}\] The first limit involving cosine simplifies. Since ah/2 aprroacheszero as h approaches zero, we are left with cos (ax) in the limit. For the second limit we use another trig identity:\[=2*\cos (ax)*\frac{a}{2}\lim_{h \rightarrow 0}\frac{\sin h}{h}*\lim_{h \rightarrow 0}\cos h\]You should recognize the limit involving sine as being equal to 1. The limit involving cosine is also equal to one. We get:\[=a \cos (ax)\]