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

(a)

Chain rule

Given g(5)=1g(5)=1 and g′(5)=−3g'(5)=-3, and f(x)=tan⁡(eg(x))f(x)=\tan\left(e^{g(x)}\right), find f′(5)f'(5).

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Hint

  1. There are three layers: tan⁡\tan, then e(⋅)e^{(\cdot)}, then g(x)g(x).

  2. Differentiate from the outside in, multiplying the derivatives.

  3. Substitute x=5x=5 and use radians.

Answer · (a)

f′(5)=−3esec⁡2(e)≈−9.81f'(5)=-3e\sec^2(e)\approx-9.81 (3 s.f.).

Radians.

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Open interactive exploration

Three layers, three factors

Step through the layers from the inside out. At x = 5 the inner value is 1.

Teal: g(5) = 1. No comparison overlay.

Start: g(5) = 1

The innermost layer: g(5)=1g(5)=1 and g′(5)=−3g'(5)=-3. This supplies the factor −3-3.

  • Running product so far
  • Value after this layer
Ideas to try
Which layer changes the sign? Which changes the size most?