Got this part ((1/(v+6))-(1/(w+6)))/(v-w). This is where I am confused. Please h

Question

Got this part ((1/(v+6))-(1/(w+6)))/(v-w).

This is where I am confused.

Please help. Need step by step.

Let the function f be given by f(x) = rac{1}{x+6}. Find the slope of the secant line that passes through the points (v,f(v)) and (w, f(w)). The slope of the secant line is: Got this part ((1/(v+6))-(1/(w+6)))/(v-w). Take the limit as w approaches v of the expression for the slope of the secant line to find the slope of the tangent line at (v,f(v)). The slope of the tangent line is This is where I am confused.

Explanation / Answer

As long the two points are given, equation of the line joining them can be found out. Secent line is one that connects two points of the function curve, while the tangent would be tangent to it at the point (there would be only one point given in that case).

Coming to the question at hand, find out the ordinates at the given values of x

for x=1, f(x)= f(1) = 1-5=-4 : the point becomes (1,-4)

for x=8, f(x) = f(8) = 64-40 = 24: the point becomes (8,24)

The line joining them would have slope = [24-(-4)]/(8-1)=28/7 = 4

Suppose the slope of tangent at any point was asked, then the function should be differentiated w.r.t.x. The value at the point could be then found by putting the values in the derivative. This is the slope of the tangent line.

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