\frac{d}{dx}Tan(X)=\frac{d}{dx} (\frac{Sin(x)}{Cos(x)})
Sin(x)=\frac{e^{ix} – e^{-ix}}{2i}
Cos(x)=\frac{e^{ix} + e^{-ix}}{2}
\frac{e^{ix} – e^{-ix}}{2i} \frac {2}{e^{ix} + e^{-ix}}
Cancel the 2s
\frac{e^{ix} – e^{-ix}}{i} \frac {1}{e^{ix} + e^{-ix}}
Multiply by
\frac{i}{i}
to clear the i from the denominator
\frac{e^{ix} – e^{-ix}}{i} \frac {1}{e^{ix} + e^{-ix}} \frac {i}{i}
i * i = -1
-i * \frac {d}{dx} ( \frac {e^{ix} – e^{-ix}} {e^{ix} + e^{-ix}} )
f(x) = e^{ix} – e^{-ix}
g(x) = e^{ix} + e^{-ix}
-i *\frac{g(x) \frac{d}{dx} f(x) – f(x) \frac{d}{dx} g(x)}{( g(x) )^2}
Start by solving the top of the equation.
-i * g(x) \frac{d}{dx} f(x) – f(x) \frac{d}{dx} g(x)
-i * [ ( e^{ix} + e^{-ix} ) ( i * e^{ix} + i * e^{-ix} ) ] – [ ( e^{ix} – e^{-ix} ) ( i * e^{ix} – i * e^{-ix} ) ]
-i * [ i * e^{2ix} + i + i + e^{-2ix} ] – [ i * e^{2ix} – i – i +e^{-2ix} ]
expand – through right side of equation
-i * [ i * e^{2ix} + (2 * i) + e^{-2ix} – i * e^{2ix} + (2 * i ) – e^{-2ix} ]
-i * [ 4 * i ]
top half = 4
bottom half of equation is
( e^{ix} + e^{-ix}) ( e^{ix} + e^{-ix})
e^{2ix} + i + i + e^{-2ix}
Final Bottom Answer is:
e^{2ix}+( 2 * i ) + e^{-2ix}
Final answer
\frac{4}{e^{2ix}+( 2 * i ) + e^{-2ix}}
Break down d/dx Tan(x)