Differentiate xsinx,x>0 w.r.t. x.
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Step-by-Step Solution
Step 1: Introduce a variable and take logarithm
To differentiate a function of the form f(x)g(x), it is often helpful to use logarithmic differentiation. We first set the given expression equal to y, and then take the natural logarithm of both sides. This allows us to use the logarithm property log(ab)=bloga to bring the exponent down.
Step 2: 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 for logy. On the right side, we will need to apply the product rule.
Step 3: Apply the product rule
We apply the product rule, which states that dxd(uv)=udxdv+vdxdu. Here, u=sinx and v=logx. We know that the derivative of logx is x1 and the derivative of sinx is cosx. Substituting these into the product rule gives us the derivative of the right side.
Step 4: Solve for dy/dx
Now we have an expression for y1dxdy. To find dxdy, we multiply both sides of the equation by y.
Step 5: Substitute y back into the equation
Finally, we substitute the original expression for y, which was xsinx, back into the equation to get the derivative in terms of x only.