How do you find the logarithmic differentiation of y = x-cos x ?
To find the logarithmic differentiation of a function, follow these steps:
Step 1: Take the natural logarithm of both sides of the equation.
ln(y) = ln(x - cos(x))
Step 2: Use the properties of logarithms to simplify the equation.
ln(y) = ln(x) - ln(cos(x))
Step 3: Differentiate both sides of the equation with respect to x.
d/dx [ln(y)] = d/dx [ln(x) - ln(cos(x))]
Step 4: Use the chain rule to differentiate ln(y).
1/y * dy/dx = 1/x - d/dx [ln(cos(x))]
Step 5: Solve for dy/dx, which is the derivative of y with respect to x.
dy/dx = y * [1/x - d/dx (ln(cos(x)))]
Now, let's simplify further.
For y = x - cos(x), we need to find dy/dx.
Step 6: Evaluate the derivative of ln(cos(x)).
d/dx [ln(cos(x))] = -tan(x)
Step 7: Substitute dy/dx, ln(cos(x)), and y into the equation.
dy/dx = (x - cos(x)) * [1/x - (-tan(x))]
dy/dx = (x - cos(x)) * (1/x + tan(x))
So, the logarithmic derivative of y = x - cos(x) is dy/dx = (x - cos(x)) * (1/x + tan(x)).