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B Chain rule for variable exponents

  1. Jan 30, 2017 #1
    I understand that when you use the chain rule you multiply the exponent by the number in front and then reduce the power by 1. So the derivative of 2x^3 = 6x^2
    I'm confused now however on how you would solve something like e^-3x, the answer turns out to be -3e^-3x

    Am I missing a rule? Why isn't it -3xe^(-3x-1) ?

    Thanks in advance
     
  2. jcsd
  3. Jan 30, 2017 #2

    Drakkith

    User Avatar

    Staff: Mentor

    As you've seen, the power rule for differentiation is: ##\frac{d}{dx}x^a = ax^{a-1}##
    But that's only the case when the variable is raised to an exponent that's a constant. In ##e^x## the exponent itself is the variable, making it an exponential function.
    The basic rule for exponential functions is: ##\frac{d}{du}e^u = e^u\frac{d}{du}u##

    For your example:
    Substituting ##u## for ##-3x##, we get ##\frac{d}{du}e^u = e^u\frac{d}{du}u##
    But ##\frac{d}{du}u## is ##\frac{d}{dx}-3x## which becomes ##-3##
    So the original equation is: ##\frac{d}{dx}e^{-3x} = -3e^{-3x}##
     
  4. Jan 30, 2017 #3

    Mark44

    Staff: Mentor

    You are using a rule (the power rule) where it isn't applicable.
    Power rule: ##\frac d {dx} x^n = nx^{n - 1}##
    In a power function, the variable is in the base. The exponent is a constant (or at least is treated as a constant as far as the differentiation is concerned.)

    ##e^{-3x}## is an exponential function, not a power function. Here the base is a constant, and the exponents is a variable or a variable expression.
    Not quite.
    ##\frac{d}{dx}e^u = e^u\frac{du}{dx}##
    Correcting the above, we have ##\frac{d}{dx}e^u = e^u\frac{du}{dx} = e^{-3x} \cdot -3 = -3e^{-3x}##
     
  5. Jan 30, 2017 #4
    Did you use the power rule to calculate the derivative of the exponent?
     
  6. Jan 30, 2017 #5

    Mark44

    Staff: Mentor

    No, I used the constant multiple rule. That is, ##\frac d{dx} kx = k##, so ##\frac d{dx}(-3x) = -3##.
     
  7. Jan 30, 2017 #6
    Ah okay , what if the exponent had an exponent? For example if it was -3x^2
     
  8. Jan 30, 2017 #7

    Mark44

    Staff: Mentor

    Use the chain rule form of the derivative of an exponential function, which @Drakkith wrote (and I modified slightly):
    ##\frac d{dx}\left(e^{-3x^2}\right) = e^{-3x^2} \frac d{dx}(-3x^2) = e^{-3x^2} \cdot -6x = -6xe^{-3x^2}##
     
  9. Jan 31, 2017 #8
    Thank you everyone
     
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