Find dy/dx if y=(ln(x))(cos(x)).
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Step-by-Step Solution
Step 1: Take logarithm on both sides
To differentiate a function of the form f(x)^{g(x)}$, we use logarithmic differentiation. We start by taking the natural logarithm on both sides of the equation.
Step 2: Apply logarithm property
Using the logarithm property ln(ab)=blna, we can simplify the right-hand side of the equation.
Step 3: Differentiate both sides with respect to x
Now, we differentiate both sides of the equation with respect to x$. On the left side, we use the chain rule. On the right side, we will use the product rule.
Step 4: Apply product rule and chain rule
Applying the product rule dxd(uv)=u′v+uv′ to the right side, where u=cosx and v=ln(lnx) We also use the chain rule for differentiating ln(lnx).
Step 5: Isolate dy/dx
To find dxdy, we multiply both sides of the equation by y.
Step 6: Substitute y back into the equation
Finally, we substitute the original expression for y ,which is(lnx)cosx, back into the equation to get the derivative in terms of x only.