Use f’( x ) = lim With h ---> 0 [f( x + h ) - f ( x )]/h to find the derivative at x for the given function. 5-x²

Answer:The derivative of the function f(x) is:                  [tex]f'(x)=-2x[/tex]Step-by-step explanation:We are given a function f(x) as:[tex]f(x)=5-x^2[/tex]We have:[tex]f(x+h)=5-(x+h)^2\\\\i.e.\\\\f(x+h)=5-(x^2+h^2+2xh)[/tex]( Since,[tex](a+b)^2=a^2+b^2+2ab[/tex] )Hence, we get:[tex]f(x+h)=5-x^2-h^2-2xh[/tex]Also, by using the definition of f'(x) i.e.[tex]f'(x)= \lim_{h \to 0} \dfrac{f(x+h)-f(x)}{h}[/tex]Hence, on putting the value in the formula:[tex]f'(x)= \lim_{h \to 0} \dfrac{5-x^2-h^2-2xh-(5-x^2)}{h}\\\\\\f'(x)=\lim_{h \to 0} \dfrac{5-x^2-h^2-2xh-5+x^2}{h}\\\\i.e.\\\\f'(x)=\lim_{h \to 0} \dfrac{-h^2-2xh}{h}\\\\f'(x)=\lim_{h \to 0} \dfrac{-h^2}{h}+\dfrac{-2xh}{h}\\\\f'(x)=\lim_{h \to 0} -h-2x\\\\i.e.\ on\ putting\ the\ limit\ we\ obtain:\\\\f'(x)=-2x[/tex]       Hence, the derivative of the function f(x) is:           [tex]f'(x)=-2x[/tex]