Completely Not Scary Differentiation
Fill in the missing words from below.
Differentiation is the act of finding the __________ __________. It tells us the gradient of the _________ at every point. The notation for the gradient function is _____.
At a certain point on the curve the tangent has the _______ gradient as the curve, so we just ____________ the value of ___ into .
The normal is at _____________ to the __________so .
At a stationary point ___________, so we set the expression ________ to 0 and ________ to find x.
The ___________ derivative identifies the _______ of stationary point when we substitute in the relevant value of x.
______________ ______________
Rules you need to know…
Chain Rule ~ function of a function
substitute t =…
Product Rule ~ function × function
Quotient Rule ~ function ÷ function
Exercise A – polynomials
Differentiate, using the chain/product/quotient rule where necessary.
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Exercise B – exponentials
Differentiate, using the chain/product/quotient rule where necessary.
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Exercise C – logarithms
Differentiate, using the chain/product/quotient rule where necessary.
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Exercise D – using your differential
1. Calculate the gradient of at x=2.
2. Find the co-ordinates of the stationary points on the curve .
3. At what point does the curve have a gradient of 2?
4. What is the equation of the tangent to the curve at the point ? (Leave your answer in terms of e.)
5. A function is such that . Find the x-coordinates of the stationary points of the graph and, by finding , decide whether each is a maximum or a minimum.
6. Find the equation of the normal to the curve at x=2.
7. Find the second derivative of .
8. Find the co-ordinates of the stationary point of and hence sketch the graph.