How do we find the derivative of #tanx# from first principal?

Answer 1

#d/(dx)tanx=sec^2x#

According to first principal, if #y=f(x)#, then
#(dy)/(dx)=Lt_(deltax->0)(f(x+deltax)-f(x))/(deltax)#
Here we have #y=f(x)=tanx#
hence #f(x+deltax)=tan(x+deltax)#
and #(dy)/(dx)=Lt_(deltax->0)(tan(x+deltax)-tanx)/(deltax)#
#=Lt_(deltax->0)(sin(x+deltax)/cos(x+deltax)-sinx/cosx)/(deltax)#
#=Lt_(deltax->0)(sin(x+deltax)cosx-cos(x+deltax)sinx)/(cosxcos(x+deltax)deltax)#
#=Lt_(deltax->0)(sin(x+deltax-x))/(cosxcos(x+deltax)deltax)#
#=Lt_(deltax->0)(sin(deltax)/(deltax) xx1/(cosxcos(x+deltax)))#
#=1xxsec^2x=sec^2x#
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Answer 2

To find the derivative of tan(x) using first principles:

  1. Start with the definition of the tangent function: tan(x) = sin(x) / cos(x).
  2. Use the definition of the derivative: f'(x) = lim(h->0) [f(x + h) - f(x)] / h.
  3. Substitute tan(x) = sin(x) / cos(x) into the derivative formula.
  4. Simplify the expression using trigonometric identities.
  5. Apply the limit as h approaches 0 to find the derivative.

The derivative of tan(x) with respect to x using first principles is:

tan'(x) = lim(h->0) [tan(x + h) - tan(x)] / h

= lim(h->0) [(sin(x + h) / cos(x + h)) - (sin(x) / cos(x))] / h

= lim(h->0) [(sin(x)cos(h) + cos(x)sin(h) - sin(x)) / (cos(x)cos(h))] / h

= lim(h->0) [(sin(x)cos(h) - sin(x)) / (cos(x)cos(h))] / h

= lim(h->0) [sin(x)(cos(h) - 1) / (cos(x)cos(h))] / h

= lim(h->0) [sin(x)(1 - cos(h)) / (cos(x)cos(h))] / h

= lim(h->0) [sin(x)(1 - cos(h)) / (cos(x)cos(x+h))] / (h/cos(x)cos(x+h))

= lim(h->0) [sin(x)(1 - cos(h)) / (cos(x)(1 - sin(x)sin(h)))] / (h/cos(x)cos(x+h))

= sin(x) / (cos^2(x))

Therefore, the derivative of tan(x) with respect to x using first principles is:

tan'(x) = sec^2(x)

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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