f(x)=sin 2x cos x

Question

f(x)=sin 2x cos x

Bytelearn · Accepted Answer

Apply Product Rule: We need to find the derivative of the function f(x) = sin(2x)cos(x). To do this, we will use the product rule, which states that the derivative of a product of two functions is the derivative of the first function times the second function plus the first function times the derivative of the second function.Define Functions: Let's denote the first function as u(x) = sin(2x) and the second function as v(x) = cos(x). According to the product rule, the derivative of f(x) with respect to x is f'(x) = u'(x)v(x) + u(x)v'(x).Find Derivative of u(x): Now we need to find the derivative of u(x) = sin(2x). The derivative of sin(2x) with respect to x is 2cos(2x), because the derivative of sin(ax) with respect to x is acos(ax), where a is a constant.Find Derivative of v(x): Next, we need to find the derivative of v(x) = cos(x). The derivative of cos(x) with respect to x is -sin(x), because the derivative of cos(x) is -sin(x).Apply Derivatives to Product Rule: Now we can apply the derivatives we found to the product rule. We have u'(x) = 2cos(2x) and v'(x) = -sin(x). Plugging these into the product rule formula, we get f'(x) = (2cos(2x))(cos(x)) + (sin(2x))(-sin(x)).Simplify Expression: Simplify the expression for f'(x) by multiplying the terms. This gives us f'(x) = 2cos(2x)cos(x) - sin(2x)sin(x).Final Answer: The expression for f'(x) is now simplified, and we have found the derivative of the function f(x) = sin(2x)cos(x). The final answer is f'(x) = 2cos(2x)cos(x) - sin(2x)sin(x).

$f(x)=\sin 2 x \cos x$

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f(x)=sin⁡2xcos⁡x f(x)=\sin 2 x \cos x f(x)=sin2xcosx

$f(x)=\sin 2 x \cos x$