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What is the derivative of # tan^-1 ((1.5)/x) - tan^-1 (0.5)#?

Answer 1

Evelyn Agard

#-6/(4x^2+9)#

#d/dx(tan^-1(1.5/x)-tan^-1(0.5))=d/dx(tan^-1(1.5x^-1)-tan^-1(0.5))#

When differentiating, #- tan^-1(0.5)# goes away as it's just a constant value. The derivative of any constant is #0.#

#d/dx(tan^-1(1.5x^-1)-tan^-1(0.5))=(1/(1+(1.5x^-1)^2))*d/dx(1.5x^-1)#

Using the Chain Rule along with the fact that

#d/dxtan^-1(x)=1/(1+x^2)#. Here, we have #1.5x^-1# as the argument of the arctangent, so instead of #x^2#, we will have #(1.5x^-1)^2# in the denominator.

#(1/(1+(1.5x^-1)^2))*d/dx(1.5x^-1)=1/(1+2.25x^-2)*-1.5x^-2#

Simplify:

#-1.5/(x^2(1+2.25/x^2))=-1.5/(x^2+(2.25cancelx^2)/cancelx^2)=-1.5/(x^2+2.25)#

#1.5=3/2# #2.25=9/4#

Getting rid of the decimals:

#-1.5/(x^2+2.25)=-(3/2)/(x^2+9/4)=-3/(2(x^2+9/4))=-3/(2x^2+9/2)=-3/((4x^2+9)/2)=-3*(2/(4x^2+9))= -6/(4x^2+9)#

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Answer 2

Christopher Allers

The derivative of ( \tan^{-1}(\frac{1.5}{x}) - \tan^{-1}(0.5) ) is ( \frac{-1.5}{x^2(1 + (\frac{1.5}{x})^2)} ).

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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.