How do you differentiate #f(x)= e^x/(e^(3-x) +2x )# using the quotient rule?

Answer 1

To differentiate a quotient, you must use the quotient rule if the function cannot be simplified.

To differentiate a quotient, use the following:

#(f(x)/g(x))' = (g(x)*f'(x) - f(x)*g'(x))/(g(x))^2#

To remember this you can use the "mnemonic":

"Low di hi minus hi di low all over low squared" where "di hi" or "di low" means to differentiate the top or the bottom respectively.

Using this strategy we get:

# = ((e^(3-x)+2x)(e^x)-(e^x)(-e^(3-x)+2))/(e^(3-x)+2x)^2#
Cleaning this up we can factor out an #e^x# and pull the negative from the fourth set of brackets in the numerator out in front to make a positive:
# = (e^x)[(e^(3-x)+2x)+(e^(3-x)-2))/(e^(3-x)+2x)^2#

And that's it! Hopefully this was clear and concise! Should you have any questions, please feel free to ask! :)

Sign up to view the whole answer

By signing up, you agree to our Terms of Service and Privacy Policy

Sign up with email
Answer 2

To differentiate ( f(x) = \frac{e^x}{e^{3-x} + 2x} ) using the quotient rule, follow these steps:

  1. Identify the numerator and denominator functions: ( u(x) = e^x ) ( v(x) = e^{3-x} + 2x )

  2. Compute the derivatives of ( u(x) ) and ( v(x) ): ( u'(x) = e^x ) ( v'(x) = -e^{3-x} + 2 )

  3. Apply the quotient rule formula: ( f'(x) = \frac{u'(x)v(x) - v'(x)u(x)}{[v(x)]^2} )

  4. Substitute the derivatives and functions into the formula: ( f'(x) = \frac{(e^x)(e^{3-x} + 2x) - (-e^{3-x} + 2)(e^x)}{[e^{3-x} + 2x]^2} )

  5. Simplify the expression: ( f'(x) = \frac{e^xe^{3-x} + 2xe^x + e^xe^{3-x} - 2e^x}{[e^{3-x} + 2x]^2} )

  6. Combine like terms in the numerator: ( f'(x) = \frac{2e^xe^{3-x} + 2xe^x - 2e^x}{[e^{3-x} + 2x]^2} )

  7. Factor out common terms: ( f'(x) = \frac{2e^x(e^{3-x} + x - 1)}{[e^{3-x} + 2x]^2} )

This is the derivative of the given function ( f(x) ) using the quotient rule.

Sign up to view the whole answer

By signing up, you agree to our Terms of Service and Privacy Policy

Sign up with email
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.

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.

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.

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.

Not the question you need?

Drag image here or click to upload

Or press Ctrl + V to paste
Answer Background
HIX Tutor
Solve ANY homework problem with a smart AI
  • 98% accuracy study help
  • Covers math, physics, chemistry, biology, and more
  • Step-by-step, in-depth guides
  • Readily available 24/7