Use logarithmic differentiation to find the derivative of the function.
y = x^(8cosx)
To find the derivative of the function y = x^(8cosx) using logarithmic differentiation, follow these step-by-step instructions:
Step 1: Take the natural logarithm (ln) of both sides of the equation to simplify the function:
ln(y) = ln(x^(8cosx))
Step 2: Apply the logarithm rules to simplify the equation. The power rule states that ln(a^b) = b ln(a):
ln(y) = (8cosx) ln(x)
Step 3: Differentiate both sides of the equation with respect to x using the chain rule. The chain rule states that d/dx (ln(u)) = (1/u) * (du/dx):
(d/dx) ln(y) = (d/dx) [(8cosx) ln(x)]
Step 4: On the left side of the equation, the derivative of ln(y) with respect to x is (1/y) * (dy/dx) using the chain rule. On the right side of the equation, the derivative of (8cosx) ln(x) with respect to x requires the product rule:
(1/y) * (dy/dx) = (d/dx) [(8cosx) ln(x)]
Step 5: Simplify the right side of the equation by using the product rule. The product rule states that d/dx (uv) = u * (dv/dx) + v * (du/dx):
(1/y) * (dy/dx) = (8cosx) * (d/dx) ln(x) + ln(x) * (d/dx) (8cosx)
Step 6: Calculate the derivatives of ln(x) and 8cosx. The derivative of ln(x) is 1/x, and the derivative of 8cosx is -8sinx:
(1/y) * (dy/dx) = (8cosx) * (1/x) + ln(x) * (-8sinx)
Step 7: Multiply both sides of the equation by y to solve for dy/dx:
dy/dx = y * [(8cosx) * (1/x) + ln(x) * (-8sinx)]
Step 8: Substitute the expression for y back into the equation. Since y = x^(8cosx), we have:
dy/dx = x^(8cosx) * [(8cosx) * (1/x) + ln(x) * (-8sinx)]
And there you have it! The derivative of the function y = x^(8cosx) using logarithmic differentiation is dy/dx = x^(8cosx) * [(8cosx) * (1/x) + ln(x) * (-8sinx)].
To find the derivative of the function using logarithmic differentiation, follow these steps:
Step 1: Take the natural logarithm of both sides of the equation.
ln(y) = ln(x^(8cosx))
Step 2: Use logarithmic properties to simplify the expression.
ln(y) = 8cosx * ln(x)
Step 3: Differentiate both sides of the equation implicitly with respect to x.
(1/y) * dy/dx = -8sinx * ln(x) + 8cosx * (1/x)
Step 4: Solve for dy/dx by multiplying both sides of the equation by y.
dy/dx = y * (-8sinx * ln(x) + 8cosx * (1/x))
Step 5: Substitute the original function back in for y.
dy/dx = x^(8cosx) * (-8sinx * ln(x) + 8cosx * (1/x))
So, the derivative of the function is dy/dx = x^(8cosx) * (-8sinx * ln(x) + 8cosx * (1/x)).
y = x^(8cosx)
lny = 8cosx lnx
1/y y' = 8(-sinx lnx + 8/x cosx)
y' = 8 x^(8cosx) (8/x cosx - sinx lnx)