Nested derivatives and increasing intervals

(a) Suppose the following facts are known about the function gg and its derivative: g(5)=1g(5)=1, g′(5)=−3g'(5)=-3. If f(x)=tan⁡(eg(x))f(x)=\tan(e^{g(x)}), find the value of f′(5)f'(5). marks

(b) Find, by differentiation, the range of values of xx for which the function y=x22x−1y=\frac{x^2}{2x-1} increases as xx increases. marks

[2009/DHS/Prelim/I/4]

Total marks: 6

Solution

If this part uses an earlier result, revisit (a) Chain rule or open all working together.

(b)

Increasing functions

Find, by differentiation, the range of values of xx for which y=x22x−1y=\dfrac{x^2}{2x-1} increases as xx increases.

Explore this idea

Hint

  1. Differentiate with the quotient rule.

  2. yy is increasing where dydx>0\frac{dy}{dx}>0.

  3. Solve the inequality, noting the denominator is a square.

Answer · (b)

x<0x<0 or x>1x>1.

Explore the idea

Open interactive exploration

Where does y = x²/(2x − 1) rise?

The upper graph is the curve with its tangent. The lower graph is the gradient. Rising stretches are shaded in both.

Position along the curvex = -1
Try an example
Test a critical position

Buttons use full-precision reviewed positions; displayed coordinates are rounded.

f(x)
-0.33333
Gradient f′(x)
0.44444
positive

For x<0 the gradient is positive: the curve rises.

Selected values are within the original drawing windows.

  • Y = x²/(2x − 1)
  • Tangent at your position
  • Gradient, and where it is positive
  • Join P to Q; the dashed legs show Δx and Δy
Key idea

yy increases exactly where the gradient graph is above the axis: x<0x<0 or x>1x>1.

Ideas to try
  1. Move across x=0x=0 and x=1x=1: what does the tangent do?
  2. What happens to the gradient near x=12x=\frac12?