SOLUTION: Need help with one more please:
Differentiate as many times as necessary to evaluate f'''(-3) if f(x) = 2 ln( absolute value of x)
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Differentiate as many times as necessary to evaluate f'''(-3) if f(x) = 2 ln( absolute value of x)
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Question 1010200: Need help with one more please:
Differentiate as many times as necessary to evaluate f'''(-3) if f(x) = 2 ln( absolute value of x) Found 2 solutions by Boreal, rothauserc:Answer by Boreal(15235) (Show Source):
You can put this solution on YOUR website! f(x)=2ln|x|
f'(x)=2/x or 2x^(-1)
f''(x)=-2x(-2) or -2/x^2
f'''(x)= 4x^(-3) or 4/x^3
f'''(-3)= -4/27
You can put this solution on YOUR website! f(x) = 2 * ln(|x|)
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Calculate the first derivative of f(x)
d/dx(f(x)) = d/dx(2 log(abs(x)))
The derivative of f(x) is f'(x):
f'(x) = d/dx(2 log(abs(x)))
Factor out constants:
f'(x) = 2 d/dx(log(abs(x)))
Using the chain rule, d/dx(log(abs(x))) = ( dlog(u))/( du) ( du)/( dx), where u = abs(x) and ( d)/( du)(log(u)) = 1/u:
f'(x) = 2 (d/dx(abs(x)))/(abs(x))
The derivative of abs(x) is x/(abs(x)):
f'(x) = (2 x/(abs(x)))/(abs(x))
Simplify the expression:
f'(x) = (2 x)/abs(x)^2
Simplifying powers, abs(x)^2 = x^2:
f'(x) = 2/x
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Calculate the second derivative of f(x)
f'(x) = 2/x
d/dx(f'(x)) = d/dx(2/x)
The second derivative of f'(x) is f''(x):
f''(x) = d/dx(2/x)
Factor out constants:
f''(x) = 2 d/dx(1/x)
Use the power rule, d/dx(x^n) = n x^(n-1), where n = -1: d/dx(1/x) = d/dx(x^(-1)) = -x^(-2):
f''(x) = 2 (-1)/(x^2)
Expand the left hand side:
f''(x) = -2/x^2
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Calculate the third derivative of f(x)
f''(x) = -2/x^2
The third derivative of f(x) is f'''(x):
f'''(x) = d/dx(-2/x^2)
Factor out constants:
f'''(x) = -2 d/dx(1/x^2)
Use the power rule, d/dx(x^n) = n x^(n-1), where n = -2: d/dx(1/x^2) = d/dx(x^(-2)) = -2 x^(-3):
f'''(x) = -2 (-2)/x^3
Simplify the expression:
f'''(x) = 4/x^3
Expand the left hand side:
f'''(x) = 4/x^3
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evaluate
f'''(-3) = 4 / (-3)^3 = -4 / 27 = -0.148148148 approx -0.15
