Can you help me find the derivative of this? Please (#cot^-1#(#sqrtt#))

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Answer 1 · 2017-11-19T03:34:55

#[ 1 / (2sqrt(t)(t + 1) } ]#

We need to differentiate #d/dt [ "arccot"(sqrt(t))]#

We use the Chain Rule to solve the problem.

Chain Rule states that .....

#d/dx[f{g(x)}] = f'{g(x)}g'(x)# to solve the problem.

Chain Rule can also be expressed as ......

#dy/(dx) = (dy)/(du)*(du)/(dx)#

So we have

#d/(dt){"arccot"(sqrt(t))} = d/(du){"arccot"(u)}(du)/(dt)#

we assume that #u = sqrt(t)#

Hence, we obtain

#d/(du){"arccot"(u)} = -1/(1 + u^2)#

#= d/(dt) {sqrt(t)} / (1 + t)#

Using Power Rule, we obtain the following:

#d/(dt) sqrt(t) = d/(dt){t ^ (1/2)}#

#= t^(-1/2)/2#

We will now write the final solution to our problem as follows:

#d/(dt){"arccot"(sqrt(t))} = 1/{2 sqrt(t) (t+1)}#

This is our final answer.