How do you differentiate #y=1/(x+ke^s)#?

Answer 1

#y=-(x+ke^s)^-2#

Rather than use quotient rule, rewrite the equation so that there's nothing in the denominator by raising everything to #-1#:

#y=(x+ke^s)^-1#

Now use the chain rule. I'm asssuming #k# and #s# are constants:

#y=-(x+ke^s)^-2#

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Answer 2

Charles Arocha

To differentiate the function ( y = \frac{1}{x + ke^s} ) with respect to ( x ), you can use the chain rule. The derivative is:

[ \frac{dy}{dx} = -\frac{1}{(x + ke^s)^2} ]

The chain rule states that if ( y = f(g(x)) ), then ( \frac{dy}{dx} = f'(g(x)) \cdot g'(x) ). In this case, ( f(u) = \frac{1}{u} ) and ( g(x) = x + ke^s ). Thus, ( f'(u) = -\frac{1}{u^2} ) and ( g'(x) = 1 ). Therefore, ( \frac{dy}{dx} = -\frac{1}{(x + ke^s)^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.