1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

How to differentiate functions with x in their exponents

  1. Jul 14, 2011 #1
    I'm not sure how to differentiate y with respect to x for:

    y = (7x^2)^[x^2]

    Any ideas?
     
  2. jcsd
  3. Jul 14, 2011 #2
    [itex] y = {(7x^2)}^{x^2} [/itex]

    Mission: To find [itex] \frac{dy}{dx} [/itex]

    We want to eliminate the problem of the exponent [itex] x^2 [/itex].
    There are several ways, but here's one neat way. Take the natural logarithm of both sides. In each step i will write in red what mathematical identity I have used to get to the expression.

    [itex] \ln(y) = \ln({(7x^2)}^{x^2}) [/itex]


    [itex] \ln(y) = x^2\ln(7x^2) [/itex] [itex] \ln{(a^b)} = b\ln(a) [/itex]

    [itex] \ln(y) = x^2\ln{({(\sqrt{7}x)}^2)} [/itex] [itex] ab^2 = {(\sqrt{a}b)}^2 [/itex]

    [itex] \ln(y) = 2x^2\ln{(\sqrt{7}x)} [/itex] [itex] \ln{(a^b)} = b\ln{a} [/itex]

    Now raise [itex] e [/itex] to the power of each side to get.

    [itex] y = e^{2x^2\ln(\sqrt{7}x)} [/itex]

    What remains is just deriving this expression, and to do so you only need to know the chain rule and product rule, and how to derive [itex] e^x [/itex].

    Ok, so lets do the remaining.

    [itex] \frac{dy}{dx} = \frac{d}{dx}e^{2x^2\ln(\sqrt{7}x)} = e^{2x^2\ln(\sqrt{7}x)}\cdot \frac{d}{dx}2x^2\ln(\sqrt{7}x) [/itex] Chain rule

    Now

    [itex] \frac{d}{dx}2x^2\ln(\sqrt{7}x) = 4x\ln(\sqrt{7}x) + 2x^2 \frac{1}{x} = 2x(\ln(7x^2) +1) [/itex] Product rule

    Thus

    [itex] \frac{dy}{dx} = 2x(\ln(7x^2) +1)e^{2x^2\ln(\sqrt{7}x)} = 2x(\ln(7x^2) +1){(7x^2)}^{x^2} [/itex]
     
    Last edited: Jul 14, 2011
  4. Jul 14, 2011 #3
    Use the chain rule with u=x^2, and v=x^2. Factor out the constant, and your answer should appear quickly.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: How to differentiate functions with x in their exponents
  1. Exponents in Functions (Replies: 15)

Loading...