how to find derative of sinx^ by first principle method

how to find derative of sinx^ by first principle method


3 Answers

879 Points
12 years ago

Dear vishal

We can approximate this value by taking a point somewhere near to P(x, f(x)), say Q(x + h, f(x + h)).

tangent delta

The value math formula is an approximation to the slope of the tangent which we require.

We can also write this slope as "change in y / change in x" or:

m = Dy/Dx

If we move Q closer and closer to P, the line PQ will get closer and closer to the tangent at P and so the slope of PQ gets closer to the slope that we want.

tangent tangent

If we let Q go all the way to touch P (i.e. h = 0), then we would have the exact slope of the tangent.

Now, math formula can be written:

math formula

So also, the slope PQ will be given by:

math formula

But we require the slope at P, so we let h → 0 (that is let h approach 0), then in effect, Q will approach P and math formula will approach the required slope.

Putting this together, we can write the slope of the tangent at P as:

math formula

This is called differentiation from first principles, (or the delta method). It gives the instantaneous rate of change of y with respect to x.

This is equivalent to the following (where before we were using h for Δx):

dy dx

You will also come across the following for delta method:



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Sagar Singh

B.Tech, IIT Delhi

Neer Varshney
76 Points
12 years ago

limh->0 sin(x).


=limh->0[sin(x)cos(h)+cos(x)sin(h)-sinx] / h.

=limh->0[sinx(cos(h)-1) +cos(x)sin(h)[/h.

=limh->0[Sinx(cos(h)-1)]/h + [cos(x)sin(h)]/h.

=0+cos(x). {(cosx-1)/x = 0. sinx/x=1}


Abhishek Panhale
16 Points
11 years ago

sinx^ what???

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