Question

For the function y = f(x) the formal definition of derivative is $f^{\prime }\left(x\right)={lim}_{\Delta x\to 0}f(X+\Delta X)-f\left(X\right).$

• A. True
• B. False

Answer 1

The correct option is B False

We know that the derivative by first principle is given by $f^{\prime }\left(x\right)={lim}_{\Delta x\to 0}\frac{f(X+\Delta X)-f\left(X\right)}{\Delta X}$
Try to recollect the fact that we conveyed two points on the graph with x coordinates as x and $x+\Delta x$. The secant joining the two points tends towards a tangent as we take the limit of $\Delta x$ as zero The slope of this tangent is nothing but the derivative of the function at that point. If you observe closely the above mentioned formula gives you the slope of that tangent only. The numerator is the difference in y coordinates and the denominator is the difference in x coordinates. Also the limit of the difference in x coordinates $\left(\Delta X\right)$ is taken as zero to make it a tangent.