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What is the derivative of this function #tan^-1(5x)#?

Answer 1

#5/(1+25x^2)#

differentiate using the #color(blue)"chain rule"#

#color(red)(|bar(ul(color(white)(a/a)color(black)(dy/dx=(dy)/(du)xx(du)/(dx))color(white)(a/a)|)))........ (A)#

#color(orange)"Reminder " color(red)(|bar(ul(color(white)(a/a)color(black)(d/dx(tan^-1 x)=1/(1+x^2))color(white)(a/a)|)))#

let #u=5xrArr(du)/(dx)=5#

and #y=tan^-1 urArr(dy)/(du)=1/(1+u^2#

Substitute these values into (A) and convert u back into x.

#rArrdy/dx=1/(1+(5x)^2) .5=5/(1+25x^2)#

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Answer 2

Adam Adkins

The derivative of the function ( \tan^{-1}(5x) ) is ( \frac{5}{1+(5x)^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.