find the derivative of:
g(x) = log[(x^3+1)^3(x^3-1)^3)]
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derivative log[(x^3+1)^3(x^3-1)^3)]
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g(x) = ln[(x^3+1)^3(x^3-1)^3)]
= 3 ln(x^3 + 1) + 3 ln (x^3 - 1)
g'(x) = 3(3x^2)/(x^3+1) + 3(3x^2)/(x^3 - 1)
= 9x^2/(x^3+1) + 9x^2/(x^3-1)
The Wolfram page assumed that you meant ln
another way is to simplify the original
RS = ln [(x^3+1)(x^3-1) ]^3
= ln (x^6 -1)^3
g(x) = ln (x^6 - 1)^3
= 3 ln(x^6 - 1)
g'(x) =3(6x^5)/(x^6 - 1) = 18x^5/(x^6 - 1)
Solve the following inequality. Then place the correct answer in the box provided. Answer in terms of an improper fraction.
3y + 5 >10
To find the derivative of the function g(x) = log[(x^3+1)^3(x^3-1)^3], we can use the chain rule and the properties of logarithms. Below are the steps to find the derivative:
Step 1: Rewrite the function using the properties of logarithms.
g(x) = 3log[(x^3+1)(x^3-1)]
Step 2: Apply the chain rule. Let u = (x^3+1)(x^3-1).
g'(x) = 3 * (1/u) * u' [Chain rule: d/dx(log(f(x))) = f'(x)/f(x)]
Step 3: Find the derivative of u using the product rule.
u' = [(x^3+1)'(x^3-1)] + [(x^3+1)(x^3-1)']
Step 4: Calculate the derivatives of x^3+1 and x^3-1.
(x^3+1)' = 3x^2
(x^3-1)' = 3x^2
Step 5: Substitute the derivative values into the expression obtained in step 3.
u' = [(3x^2)(x^3-1)] + [(x^3+1)(3x^2)]
Step 6: Simplify the expression.
u' = 3x^5 - 3x^2 + 3x^5 + 3x^2
u' = 6x^5
Step 7: Substitute the value of u' into the expression obtained in step 2.
g'(x) = 3 * (1/u) * 6x^5
Step 8: Simplify the expression.
g'(x) = 18x^5 / [(x^3+1)(x^3-1)]
Therefore, the derivative of g(x) = log[(x^3+1)^3(x^3-1)^3] is g'(x) = 18x^5 / [(x^3+1)(x^3-1)].