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Let F(x)=f(x)g(x)h(x) for all real x ,w ...

Let `F(x)=f(x)g(x)h(x)` for all real `x ,w h e r ef(x),g(x),a n dh(x)` are differentiable functions. At some point `x_0,F^(prime)(x_0)=21 F(x_0),f^(prime)(x_0)4f(x_0),g^(prime)(x_0)=-7g(x_0),` then the value of `g^(prime)(1)` is ________

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`F(x)=f(x)g(x)h(x)AA x in R`
f(x),g(x), h(x) are differentiable functions. Then
`F'(x)=f'(x)g(x)h(x)+f(x)g'(x)h(x)+f(x)g(x)h'(x)`
`"At "x=x_(0),`
`F'(x_(0))=f(x_(0))g(x_(0))h(x_(0))g'(x_(0))h(x_(0))+f(x_(0))g(x_(0))h'(x_(0)).`
`"Using the given values of "F'(x_(0)),f'(x_(0)),g'(x_(0)), and h'(x_(0))." we get "`
`21F(x_(0))=4f(x_(0))g(x_(0))h(x_(0))-7f(x_(0))g(x_(0))h(x_(0))+kh(x_(0))f(x_(0))g(x_(0))`
`"or "21=4-7+k" "[because F(x_(0))=f(x_(0))g(x_(0))h(x_(0))]`
`"or "k=24`
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CENGAGE-DIFFERENTIATION-Exercise (Numerical)
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