Find the derivative of f(x)=log(7-2x)?
2017-09-27 3:35 pm
I was wondering if someone could explain how to go about finding the derivative of logarithms such as the example above. I know I have to use the chain rule, but for some reason, I m having a hard time getting over the bump with this type of problem. Thanks for any help!
回答 (6)
2017-09-27 4:07 pm
f(x) = ln(7 - 2x)
(I'm assuming the natural log here, but any other logarithm base simply differs by a multiplicative constant).
Let u = 7 - 2x, then f = ln(u), and by the chain rule, df/dx = df/du*du/dx
du/dx -2
df/du = 1/u
So:
df/dx = -2/u = -2/(7 - 2x) = 2/(2x - 7)
(I'm assuming the natural log here, but any other logarithm base simply differs by a multiplicative constant).
Let u = 7 - 2x, then f = ln(u), and by the chain rule, df/dx = df/du*du/dx
du/dx -2
df/du = 1/u
So:
df/dx = -2/u = -2/(7 - 2x) = 2/(2x - 7)
2017-09-27 5:22 pm
Assuming that it is a decimal log (base 10), we have:
f'(x) = [log(7 - 2x)]' * (7 - 2x)' = 1/[(7 - 2x) * ln (10)] * (-2) = -2/[(7 - 2x) * ln (10)].
Or, alternatively: f'(x) = 2/[(2x - 7) * ln (10)] (we multiplied both the numerator and the denominator by -1).
P.s.: If the base of the logarithm equals e, the derivative simplifies to:
f'(x) = 2/[(2x - 7) * ln (e)] = 2/[(2x - 7) * 1] = 2/[(2x - 7)].
f'(x) = [log(7 - 2x)]' * (7 - 2x)' = 1/[(7 - 2x) * ln (10)] * (-2) = -2/[(7 - 2x) * ln (10)].
Or, alternatively: f'(x) = 2/[(2x - 7) * ln (10)] (we multiplied both the numerator and the denominator by -1).
P.s.: If the base of the logarithm equals e, the derivative simplifies to:
f'(x) = 2/[(2x - 7) * ln (e)] = 2/[(2x - 7) * 1] = 2/[(2x - 7)].
2017-09-27 5:13 pm
Let y = log u
u = 7 - 2x
du/dx = - 2
dy/du = 1/u
dy/dx= - 2 / u = - 2 / [ 7 - 2x ]
u = 7 - 2x
du/dx = - 2
dy/du = 1/u
dy/dx= - 2 / u = - 2 / [ 7 - 2x ]
2017-09-27 3:42 pm
-2/(7-2x). It is the chain rule. the derivative of log(x) is 1/x, so then by the chain rule the derivative of (im going to cal g(x)=7-2x) log(g(x)) is (1/g(x)) times the derivative of g(x). So here 1/gx) is 1/(7-2x), and the derivative of g(x) is -2, so the derivative of f(x)=log(g(x))=-2/(7-2x)
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