Given: A function f(x)
[tex]f(x)=x^2+2x[/tex]
Required: To verify the derivative of the given function is
[tex]f^{\prime}(x)=2x+2[/tex]
by using the method of Limits.
Explanation: The derivative of a function can be calculated by using limits-
[tex]f^{\prime}(x)=\lim_{h\to0}\frac{f(x+h)-f(x)}{h}[/tex]
Here we have
[tex]\begin{gathered} f(x)=x^2+2x \\ f(x+h)=(x+h)^2+2(x+h) \\ =x^2+h^2+2xh+2x+2h \end{gathered}[/tex]
Putting these values in Limit we get
[tex]\begin{gathered} f^{\prime}(x)=\lim_{h\to0}\frac{x^2+h^2+2xh+2x+2h-x^2-2x}{h} \\ =\lim_{h\to0}\frac{h^2+2xh+2h}{h} \\ =\lim_{h\to0}\frac{h(h+2x+2)}{h} \\ =2x+2 \end{gathered}[/tex]
Hence the result is verified.
Final Answer: The derivative of the given function is
[tex]f^{\prime}(x)=2x+2[/tex]
