Differentiate #xsinx# using first principles?
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# d/dx xsinx = x cosx +sinx #
Let us define:
Using the limit definition of the derivative, we compute the derivative using:
So we only need to calculate the limit:
We get:
With these results, we get the result:
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To differentiate ( x\sin(x) ) using first principles:

Start with the definition of the derivative: [ f'(x) = \lim_{h \to 0} \frac{f(x + h)  f(x)}{h} ]

Substitute ( f(x) = x\sin(x) ) into the definition: [ f'(x) = \lim_{h \to 0} \frac{(x + h)\sin(x + h)  x\sin(x)}{h} ]

Expand and simplify the expression: [ f'(x) = \lim_{h \to 0} \frac{x\sin(x) + h\sin(x) + x\sin(h) + h\sin(h)  x\sin(x)}{h} ] [ f'(x) = \lim_{h \to 0} \frac{h\sin(x) + x\sin(h) + h\sin(h)}{h} ] [ f'(x) = \lim_{h \to 0} \left(\sin(x) + \frac{x\sin(h) + h\sin(h)}{h}\right) ]

Use the limit properties to separate the limit: [ f'(x) = \sin(x) + \lim_{h \to 0} \left(\frac{x\sin(h) + h\sin(h)}{h}\right) ]

Recognize that ( \lim_{h \to 0} \frac{x\sin(h)}{h} = x ) and ( \lim_{h \to 0} \frac{h\sin(h)}{h} = 0 ): [ f'(x) = \sin(x) + x ]
Therefore, the derivative of ( x\sin(x) ) with respect to ( x ) using first principles is ( \sin(x) + x ).
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When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
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