Question

Find dy/dx (the derivative ) of f(x) = 4x^2 using limits definition of the derivitive (of limit of the DQ when h> 0Formulas neededa) Equation of a line in slopeyintercept form y = f(x) = mx + bb) Equation of a line slopepoint y y1 = m (x x1)

Answer 1

Given

`f(x)=4x^2`

Find

derivative of the function

Step-by-step explanation

By limit definition ,

`(dy)/(dx)=\lim_(h\to0)(f(x+h)-f(x))/(h)`

so, we have

`\begin{gathered} (dy)/(dx)=\lim_(h\to0)(\lbrace4\left(x+h\right)^2-4x^2\rbrace)/(h) \\ (dy)/(dx)=\lim_(h\to0)(\lbrace4(x^2+h^2+2xh)^-4x^2\rbrace)/(h) \\ (dy)/(dx)=\lim_(h\to0)(\lbrace4x^2+4h^2+8xh-4x^2\rbrace)/(h) \\ (dy)/(dx)=\lim_(h\to0)(8xh+4h^2)/(h) \\ (dy)/(dx)=\lim_(h\to0)8x+4h \\ (dy)/(dx)=8x \end{gathered}`

Final Answer

The derivative of the function is 8x