Hey there! I need help with this problem, please!

Question

Find the derivative of the following function: y=sqrt(x)/e^x

And if you can, could you explain the solution step by step, so I can understand how to solve this type of problems. Thank you!

Answer 1

Hey Almaz,

I hope this helps. Basically the same thing as the other answers plus a little color coating. The Quotient rule/product rule can both apply, but I showed you the quotient rule way and a little way to remember it as well.

Answer 2

This question requires the quotient rule. Recall that, if your numerator function is f(x) and your denominator function is g(x), the derivative of the quotient f(x)/g(x) is (d/dx)(f/g)=[gf'-fg']/g^2. In this problem, the numerator f(x) is √x=x^1/2, so f'(x)=(1/2)x^-1/2 by the power rule, and the denominator g(x) is g(x)=e^x, so g'(x)=e^x. Thus, the quotient rule tells us that (d/dx)(x^1/2/e^x)=[(e^x)(1/2)(x^-1/2)-(x^1/2)(e^x)]/(e^x)²=(1/2)e^x[x^-1/2-2x^1/2]/[(e^x)(e^x)]=[x^-1/2-2x^1/2]/(2e^x), and since x^-1/2-2x^1/2=(1/√x)-2(√x)×[(√x)/(√x)]=(1/√x)-(2x/√x), we see

(d/dx)(x^1/2/e^x)=(1-2x)/[2(√x)e^x]. This last step was meant to eliminate the "fraction within a fraction" issue (well, it's not that much of a problem, to be honest) that makes the answer messy.

Answer 3

You can either use the quotient formula or the product formula to get this answer, the latter my be a little easier. Remember that 1/e^x = e^-x and sqrt(x) = x^1/2

I will assume you can get the derivatives of the individual terms.

Hence the derivative is

x^1/2(-e^-x) + e^-x(1/2)x^-1/2.

You can simplify this a little with some algebra, but the answer remains the same.