find the derivative
f(x)= 8ln(x^5-2x)
well, if g=ln u
then g'=1/u du/x
so here, that translates to
f'=8/(x^5-2x) * (5x^4-2)
To find the derivative of the function f(x) = 8ln(x^5 - 2x), we can use the chain rule and the logarithmic differentiation technique. Here's how to do it:
Step 1: Apply the logarithmic differentiation technique.
- Take the natural logarithm of both sides of the equation: ln(f(x)) = ln(8ln(x^5 - 2x)).
- Express the given function in its logarithmic form: ln(f(x)) = ln(8) + ln(ln(x^5 - 2x)).
Step 2: Differentiate both sides of the equation using the chain rule.
- Differentiate ln(f(x)) with respect to x: (1/f(x)) * f'(x).
- Differentiate ln(8) with respect to x: 0 (since 8 is a constant).
- Differentiate ln(ln(x^5 - 2x)) using the chain rule: (1/ln(x^5 - 2x)) * [d/dx(ln(x^5 - 2x))].
Step 3: Compute d/dx(ln(x^5 - 2x)).
To differentiate ln(x^5 - 2x), we need to apply the chain rule.
- Let u = x^5 - 2x.
- Differentiate u with respect to x as du/dx = 5x^4 - 2.
Step 4: Substitute the values back into the equation.
We have:
(1/f(x)) * f'(x) = 0 + (1/ln(x^5 - 2x)) * (5x^4 - 2).
Step 5: Simplify the expression.
Multiply through by f(x) to get rid of the fraction:
f'(x) = f(x) * [(5x^4 - 2)/[(x^5 - 2x) * ln(x^5 - 2x)]].
Substituting f(x) = 8ln(x^5 - 2x) into the equation:
f'(x) = 8ln(x^5 - 2x) * [(5x^4 - 2)/[(x^5 - 2x) * ln(x^5 - 2x)]].
Simplifying the expression further as much as possible:
f'(x) = 8 * (5x^4 - 2)/(x^5 - 2x).
Therefore, the derivative of f(x) = 8ln(x^5 - 2x) is f'(x) = 8 * (5x^4 - 2)/(x^5 - 2x).