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Use the definition of the derivative to prove the following special case of the product rule

ddx(x2f(x))=2xf(x)+x2f'(x)

Short Answer

Expert verified

We proved the special case of product function using the definition of the derivative

Step by step solution

01

Given information

We are given a functionddx(x2f(x))=2xf(x)+x2f'(x)

02

Find the derivative

Consider g(x)=x2f(x)

Using the definition of derivative we get,

limh0g(x+h)-g(x)hlimh0(x+h)2f(x+h)-x2f(x)hlimh0(x2+2xh+h2)f(x+h)-x2f(x)hlimh0x2(f(x+h)-f(x))h+limh0h(2x+h)f(x+h)h=x2f'(x)+2xf(x)

Hence proved

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