use logarithmic differentiation to differentiate the function f(x)=2^x(x^2+2)^3(x^3-3)^7/(x^2+4)^1/2
remember that log(product) = sum of logs of factors
let y = 2^x(x^2+2)^3(x^3-3)^7/(x^2+4)^1/2
take ln of both sides
ln y = ln 2^x + ln (x^2+2)^3 + ln (x^3 - 3)^7 - ln (x^2+4)^(1/2)
ln y = x (ln2) + 3 ln(x^2+2) + 7 ln(x^3-3) - (1/2) ln(x^2 + 4)
y' /y = ln2 + 6x/(x^2 + 2) + 21x^2/(x^3-3) - x/(x^2+4)
y' = dy/dx = y [ln2 + 6x/(x^2 + 2) + 21x^2/(x^3-3) - x/(x^2+4) ]
don't know what kind of simplification you need, but that would be just careful algebra
To use logarithmic differentiation to differentiate the function f(x) = 2^x(x^2 + 2)^3(x^3 - 3)^7/(x^2 + 4)^(1/2), follow these steps:
Step 1: Take the natural logarithm of both sides of the equation:
ln(f(x)) = ln(2^x(x^2 + 2)^3(x^3 - 3)^7/(x^2 + 4)^(1/2))
Step 2: Use logarithmic properties to simplify the equation. Remember that ln(ab) = ln(a) + ln(b), and ln(a^b) = b * ln(a). Apply these properties to each term in the equation:
ln(f(x)) = ln(2^x) + ln((x^2 + 2)^3) + ln((x^3 - 3)^7) - (1/2) * ln((x^2 + 4))
Step 3: Apply the power rule of logarithms and differentiate each term on the right side of the equation related to x:
ln(f(x)) = x * ln(2) + 3 * ln(x^2 + 2) + 7 * ln(x^3 - 3) - (1/2) * ln(x^2 + 4)
Taking the derivative of both sides of the equation with respect to x yields:
1/f(x) * f'(x) = ln(2) + (3 * 2x)/(x^2 + 2) + (7 * 3x^2)/(x^3 - 3) - (1/2) * (2x)/(x^2 + 4)
Step 4: Solve the equation for f'(x). Multiply both sides of the equation by f(x):
f'(x) = f(x)[ln(2) + (3 * 2x)/(x^2 + 2) + (7 * 3x^2)/(x^3 - 3) - (1/2) * (2x)/(x^2 + 4)]
Finally, substitute the expression for f(x) back into the equation to obtain the derivative in its final form.