The instantaneous rate of change of a function let it be f(x) is given by it's first Derivative i.e f'(x) and at a particular point like x = a , it's given by f'(a) . But let's recall a basic formula first :
- [tex]{\boxed{\bf{\dfrac{d}{dx}(x^{n})=n{x}^{n-1}}}}[/tex]
So , now here ;
[tex]{:\implies \quad \sf f(x)=-4x^{2}+2x}[/tex]
Differentiating both sides w.r.t.x ;
[tex]{:\implies \quad \sf f^{\prime}(x)=-4\dfrac{d}{dx}(x^2)+2\dfrac{dx}{dx}}[/tex]
[tex]{:\implies \quad \sf f^{\prime}(x)=-4\cdot 2 \cdot x+2}[/tex]
[tex]{:\implies \quad \sf f^{\prime}(x)=-8x+2}[/tex]
Now , at x = 3 ;
[tex]{:\implies \quad \sf f^{\prime}(3)=-8(3)+2}[/tex]
[tex]{:\implies \quad \sf f^{\prime}(3)=-24+2}[/tex]
[tex]{:\implies \quad \bf \therefore \quad \underline{\underline{f^{\prime}(3)=-22}}}[/tex]
Hence , the required answer is -22
