Find f′(x) if f′(x)=(sinx)sinx for all 0<x<π.
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Step-by-Step Solution
Step 1: Take the natural logarithm of both sides
To differentiate a function of the form u(x)v(x), it is often helpful to use logarithmic differentiation. We start by taking the natural logarithm of 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=sinx and v=ln(sinx). The derivative of sinx is cosx, and the derivative of ln(sinx) is sinx1⋅cosx by the chain rule.
Step 5: Simplify the expression
We simplify the right-hand side by canceling out sinx in the second term.
Step 6: Solve for f′(x)
Finally, we multiply both sides by f(x) and substitute back f(x)=(sinx)sinx. We can also factor out cosx from the expression.