Further differentiation

1. Differentiate the following functions

(a)

 Hint

we apply chain rule

 Solutions

(b)

 Hint

we apply product rule

 Solutions

 Hint

we first apply chain rule on , then apply quotient rule:

 Solutions

(d) where is a positive constant greater than 1 .

 Hint

We can use chain rule:
function 1 is:
function 2 is:

Remember the formula for differentiation of log of any base:

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(a) Show from first principles that for the curve , where is in radians,

You may use the formula for without proof and assume that as ,

 Hint

Start with expanding using the compound angle formulae. will be , the very small changes of angle in radian.

 Solutions

From first principle: .

(b) Identify where in the proof this assumption that is in radians is needed.

 Solutions

Based on small angle approximation, need to be in radian, because.
It can only be true when is in radian, it is not true when is in angle.

(a) Find an expression in terms of and for the gradient of the curve

 Hint

Differentiation of Implicit function, together with chain rule and product rule. For example,

 Solutions

(b) The diagram shows the curve where .

Find the values of for which the tangent to the curve is vertical.

 Hint

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If tangent is vertical, then the denominator of the derivative (gradient) function will be 0. That means:

Combined with original curve, we can have following equation set:

Explain what happens to the gradient when .

 Hint

Substitute into original function, and rearranged, then see what kind of function it is

 Solutions

when , the curve of the original function contains two parallel lines with gradient of

4. In this question you must show detailed reasoning

Show that the curve has one turning point only, and give the coordinates of this point.

 Hint

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5. In this question you must show detailed reasoning

A curve has

(a) Show that the turning points of the curve occur at the points for which .

 Hint

Get the differentiation (gradient) function, then set it to 0. Then rearrange and simplify.

 Solutions

(b) Find the equation of the normal to the curve at the point for which .

 Hint

Find the gradient at , then the normal will be negative reciprocal

 Solutions