If y=sin(2x)/x and x =Pi and ex=.25 find the differentaite
Solution;
Given;
x=π
ex=0.25e^x=0.25ex=0.25
y=sin(2x)xy=\frac{sin(2x)}{x}y=xsin(2x)
ln(ex)=ln(0.25)ln(e^x)=ln(0.25)ln(ex)=ln(0.25)
x=ln(0.25)x=ln(0.25)x=ln(0.25)
Apply quotient rule;
dydx=2xcos(2x)−sin(2x)x2\frac{dy}{dx}=\frac{2xcos(2x)-sin(2x)}{x^2}dxdy=x22xcos(2x)−sin(2x)
At x=π;
dydx=2πcos(2π)−sin(2π)π2\frac{dy}{dx}=\frac{2πcos(2π)-sin(2π)}{π^2}dxdy=π22πcos(2π)−sin(2π) =2π−0π2=2π\frac{2π-0}{π^2}=\frac2ππ22π−0=π2
At x=ln(0.25)
dydx=2ln0.25cos(2×ln0.25)−sin(2×ln0.25)(ln0.25)2\frac{dy}{dx}=\frac{2ln0.25cos(2×ln0.25)-sin(2×ln0.25)}{(ln0.25)^2}dxdy=(ln0.25)22ln0.25cos(2×ln0.25)−sin(2×ln0.25)
dydx=−1.415\frac{dy}{dx}=-1.415dxdy=−1.415
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