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Find the derivative of cos2x, by using the first principle of derivatives.

Answer
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Hint: We recall the first principle of derivative. We assume a small change in x as δx and its corresponding change in y=f(x) as δy. We find the average rate of change as δyδx=f(x+δx)f(x)δx . We take limit δx0 to find the instantaneous rate of change as derivative of f(x).

Complete step-by-step solution:
We are given the function f(x)=cos2x in the question. Let us havey=cos2x. Let δx be a very small change in x and the corresponding change in y be δy. So we have;
y+δy=cos2(x+δx)
We subtract y both sides of the above equation to have;
y+δyy=cos2(x+δx)yy+δyy=cos2(x+δx)cos2xδy=cos2(x+δx)cos2x
We divide δx both sides of the above step to have;
δyδx=cos2(x+δx)cos2xδx
We take limit δx0 both sides of the above step to have;
limδx0δyδx=limδx0cos2(x+δx)cos2xδx
We use the trigonometric identity cos2Bcos2A=sin(A+B)sin(AB) for A=x,B=x+δx in the above step to have;
limδx0δyδx=limδx0sin(x+δx+x)sin(xδxx)δxlimδx0δyδx=limδx0sin(2x+δx)sin(δx)δxlimδx0δyδx=limδx0sin(2x+δx)sin(δx)δx
We use law of product if limits in the right hand side of the above step to have;
limδx0δyδx=limδx0sin(2x+δx)limδx0sin(δx)δx
We use the standard limit limxosinxx=1 for x=δx in the right hand side of the above step to have;
 limδx0δyδx=limδx0sin(2x+δx)1limδx0δyδx=limδx0sin(2x+δx)limδx0δyδx=sin2x
We use the double angle formula sin2θ=2sinθcosθ for θ=x in the right hand side of the above step to have;
limδx0δyδx=2sinxcos
We know from first principle of derivative that limδx0δyδx=dydx. So we have
dydx=2sinxcosxddx(cos2x)=2sinxcosx

Note: We can use chain rule to directly find the derivative of cos2x. If composite function is defined as y=u(x),u=f(x) then the chain rule is given as dydx=dydududx. We can also use the first principle for derivative with a very small change h as ddxf(x)=limh0f(x+h)f(h)h. The derivative of the function at particular points geometrically gives the slope of the tangent to the curve of the function. The first principle is also known as the delta method.


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