f(x)=6(sin(x))^x, find f'(1).
and
f(x)=3x^ln(x), find f'(9).
i tried these so many times and it keeps telling me that my answers are wrong :/
for the first one i did f'(1) = 6sin(1) * {ln[sin(1)] + cot(1)}., is that correct, when i put it in my calculator and put my answer it, it says im wrong
To find the derivative of f(x) = 6(sin(x))^x and evaluate it at x = 1, we can use the logarithmic differentiation method.
Step 1: Take the natural logarithm of both sides of the equation f(x) = 6(sin(x))^x:
ln(f(x)) = ln(6(sin(x))^x)
Step 2: Apply the logarithm rules to simplify the equation. Using the property ln(a^b) = b * ln(a) and the logarithm identity ln(ab) = ln(a) + ln(b), we get:
ln(f(x)) = x * ln(6(sin(x)))
Step 3: Differentiate both sides of the equation with respect to x using the chain rule:
(1 / f(x)) * f'(x) = ln(6(sin(x))) + x * (1 / (6(sin(x)))) * cos(x)
Step 4: Solve for f'(x):
f'(x) = f(x) * [ln(6(sin(x))) + x * (1 / (6(sin(x)))) * cos(x)]
Step 5: Evaluate f'(1):
Plug in x = 1 into the equation for f'(x):
f'(1) = f(1) * [ln(6(sin(1))) + 1 * (1 / (6(sin(1)))) * cos(1)]
Now we can evaluate the expression numerically using a calculator:
f'(1) ≈ 6(sin(1))^1 * [ln(6sin(1)) + 1 * (1 / (6sin(1))) * cos(1)]
However, please note that the exact value of f'(1) may require you to use a calculator with higher precision or to use numerical approximation methods due to the trigonometric and logarithmic functions involved.
As for the second question:
To find the derivative of f(x) = 3x^ln(x) and evaluate it at x = 9, we can use logarithmic differentiation as well.
Step 1: Take the natural logarithm of both sides of the equation f(x) = 3x^ln(x):
ln(f(x)) = ln(3x^ln(x))
Step 2: Simplify the equation using the logarithm rules:
ln(f(x)) = ln(3) + ln(x^ln(x))
Step 3: Use the properties of logarithms to simplify further:
ln(f(x)) = ln(3) + ln(x) * ln(x)
Step 4: Differentiate both sides of the equation with respect to x using the chain rule:
(1 / f(x)) * f'(x) = (1 / x) * ln(x) + (ln(x) * 1) * (1 / x)
Step 5: Solve for f'(x):
f'(x) = f(x) * [(1 / x) * ln(x) + (ln(x) / x)]
= 3 * x^ln(x) * [(1 / x) * ln(x) + (ln(x) / x)]
Step 6: Evaluate f'(9):
Plug in x = 9 into the equation for f'(x):
f'(9) = 3 * 9^ln(9) * [(1 / 9) * ln(9) + (ln(9) / 9)]
Now you can evaluate this expression numerically using a calculator.