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Derivative of power function

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  • #1
Qube
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Derivative of exponential function

Homework Statement



f(x) = 4^x + e^(2tanx)

find f'(0)

Homework Equations



I used log differentiation.

y = 4^x + e^(2tanx)

lny = xln4 + 2tanx(lne) = xln4 + 2tanx

The Attempt at a Solution



y'/y = ln4 + 2sec^2(x)

Solving for y I get y = 2.

Plugging in x = 0 I get:

y'/2 = ln4 +2sec^(0) = ln4 + 2(1)

y' = 2(ln4+2)

This however, isn't the answer.
 
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Answers and Replies

  • #2
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Homework Statement



f(x) = 4^x + e^(2tanx)

find f'(0)

Homework Equations



I used log differentiation.

y = 4^x + e^(2tanx)

lny = xln4 + 2tanx(lne) = xln4 + 2tanx

The Attempt at a Solution



y'/y = ln4 + 2sec^2(x)

Solving for y I get y = 2.
No - don't do this yet. Solve for y' algebraically.

There's a difference between y' and y'(0). The first is a function, and the second is a number. Your need the function.
Plugging in x = 0 I get:

y'/2 = ln4 +2sec^(0) = ln4 + 2(1)

y' = 2(ln4+2)

This however, isn't the answer.
 
  • #3
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Btw, your title is misleading. 4x is NOT a power function. It's an exponential function.

In a power function, the variable is in the base, and the exponent is constant. E.g., x3, y5, and so on are power functions.
 
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  • #4
Qube
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No - don't do this yet. Solve for y' algebraically.

There's a difference between y' and y'(0). The first is a function, and the second is a number. Your need the function.
Alright.

y' = y(ln4 + 2sec^2(x))

How do I proceed? Do I plug in y =2? If so I still get the same answer if I plug in x = 0.
 
  • #5
LCKurtz
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Homework Statement



f(x) = 4^x + e^(2tanx)

find f'(0)

Homework Equations



I used log differentiation.

y = 4^x + e^(2tanx)

lny = xln4 + 2tanx(lne) = xln4 + 2tanx
This step in incorrect. ##\ln(a+b) \ne \ln a + \ln b##.
 
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  • #6
Qube
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I got it now! Thanks! I should probably review my log rules. It's amazing that after two years of calculus and a 5 on both AP Calc AB and BC I still make the simplest of pre-calculus mistakes!
 
  • #7
Ray Vickson
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Homework Statement



f(x) = 4^x + e^(2tanx)

find f'(0)

Homework Equations



I used log differentiation.

y = 4^x + e^(2tanx)

lny = xln4 + 2tanx(lne) = xln4 + 2tanx

The Attempt at a Solution



y'/y = ln4 + 2sec^2(x)

Solving for y I get y = 2.

Plugging in x = 0 I get:

y'/2 = ln4 +2sec^(0) = ln4 + 2(1)

y' = 2(ln4+2)

This however, isn't the answer.
Your second line essentially says that
[tex]\log(a+b) = \log(a) + \log(b)\;\leftarrow \text{ false!}[/tex]
What IS true is ## \log(a\cdot b) = \log(a) + \log(b)##.
 
  • #8
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This step in incorrect. ##\ln(a+b) \ne \ln a + \ln b##.
I glossed right over that in the OP's work. Oh, well.
 

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