Differentials. Consider the following function and express the relationship between a small change in x and the corresponding change in y in the form dy=f '(x)dx f(x)= x+4/4-x

Question

Dfferentials. Consider the following function and express the relationship between a small change in x and the corresponding change in y in the form dy=f '(x)dx f(x)= x+4/4-x

Answer 1

y=f(x)=(4+x)/(4-x).

If x increases by a small amount h then y increases by a small amount k. dy/dx approximates to k/h.

k=f(x+h)-f(x)=(4+x+h)/(4-x-h)-(4+x)/(4-x)=

((4+x+h)(4-x)-(4+x)(4-x-h))/((4-x-h)(4-x))=

(16-x²+h(4-x)-(16-x²-h(4+x))/((4-x)²-h(4-x))=

(4h-xh+4h+xh)/((4-x)²-h(4-x))=

8h/((4-x)²-h(4-x)). In the denominator the term h(4-x) is small compared to (4-x)², so we can ignore it.

Therefore k=8h/(4-x)² and k/h=8/(4-x)².

The derivative of f(x)=f'(x)=(4-x+(4+x))/(4-x)²=8/(4-x)². This is the same as k/h=dy/dx.