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QuestionMathematics
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Step 1:I will solve the following calculus problem:
Find the derivative of the function $$f(x) = x^3 \sin(x)$$.
Step 2:: Recall the product rule for differentiation.
In this case, let $$g(x) = x^3$$ and $$h(x) = \sin(x)$$.
Step 3:: Find the derivatives of g(x) and h(x).
h'(x) = \cos(x)
We know that: and
Step 4:: Plug these expressions into the product rule formula from Step 1:
f'(x) = (3x^2)(\sin(x)) + (x^3)(\cos(x))
Step 5:: Simplify the expression:
f'(x) = 3x^2 \sin(x) + x^3 \cos(x)
Final Answer
The derivative of the function f(x) = x^3 \sin(x) is f'(x) = 3x^2 \sin(x) + x^3 \cos(x).
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