Question

Given , find the value of   when x=8, writing your answer in the form  where a is a rational number.

Solution

We are given;

We are required to find .

Gradient (slope) of the curve is the derivative of equation of the curve. Hence gradient of curve  with respect to  is:

Therefore;

Rule for differentiation is of  is:

Rule for differentiation is of  is:

Rule for differentiation is of  is:

We need to find the gradient of the curve when x=8.

Gradient (slope) of the curve at the particular point is the derivative of equation of the curve at that  articular point.

Gradient (slope)  of the curve  at a particular point  can be found by substituting x- coordinates of that point in the expression for gradient of the curve;

We have already found that;

Substitute  in derivative of the equation of the curve.

Since ;

If we need a rational number in the denominator of a fraction, we need to follow procedure of  “denominator rationalization” as given below.

ü If the denominator is of the form  then multiply both numerator and denominator by . 

ü If the denominator is of the form  then multiply both numerator and denominator by  .

ü If the denominator is of the form  then multiply both numerator and denominator by  .