MATH1011 Chap.9 Differentiation Rules and Techniques
Differentiation Rules and Techniques
Every rule in this chapter is one limit in disguise
Fix a point on a curve, step a distance h along the x axis, and join the two points with a straight line. The gradient of that line is a fraction, the change in output over the change in input, and it reports the average rate of change across the step. Now let h shrink towards 0. If the fraction settles on a number, that number is the derivative at the fixed point.
Every formula that follows is a labour saving device for exactly that calculation, which is why a formula you half remember and never derived tends to collapse on the first expression you have not seen before.
Reading the answer is half the value of computing it. A positive derivative says the output is climbing as the input grows there, a negative one says it is falling, and 0 says the tangent lies flat.
The size is a claim you can test: a derivative of 12 says that near that point about twelve units of output arrive per unit of input, so take a small step either side, divide the change in output by the change in input, and see whether the two agree to three or four figures. That thirty second check catches a dropped factor faster than rereading your own algebra does.
The units come along too: if y is a cost in dollars and x a number of items, dy/dx is dollars per item, and a bare number has answered only part of the question.
Ask which operation you would perform last
Faced with a long expression, the useful question is not which formulas you can recite but which single operation finishes the job when you evaluate the thing at a number.
That final operation names the rule, and it is visible in the shape of the expression rather than in its vocabulary. If the last act is a multiplication of two varying factors, the product rule owns the expression. If it is a division, the quotient rule owns it.
If it is one function wrapped around a whole expression, such as a power, a logarithm or a sine, the chain rule owns it, and you peel the layers from the outside inwards, one factor for each. Two expressions can be described in almost the same words while their last operations differ, and then their derivatives differ too.
Two habits save marks here.
Keep the entire top of a quotient inside a bracket until after the subtraction is written down, so the minus sign reaches every term of the second product. And resist tidying the expression into a form you find prettier before you differentiate: expanding a cube by hand to dodge the chain rule introduces more arithmetic slips than the chain rule ever would.
The list of basic derivatives underneath all of this is short, and it is worth being able to write it out cold: a constant power drops its exponent in front, the natural exponential returns itself, a general exponential brings down the natural logarithm of its base, the natural logarithm returns one over x, and sine and cosine trade places with a minus sign arriving on the cosine line.
Two cues that send you down a side road
Most expressions give way to the three main rules.
Two situations do not, and each one announces itself in the question. The first cue is a relation between x and y that you cannot rearrange into y on its own, because y turns up squared, or inside a logarithm, or multiplied by x. Differentiate it as it stands, treating y as an unnamed function of x, so a dy/dx is born wherever something containing y was differentiated.
Then gather those terms on one side, take dy/dx outside a bracket, and divide. Any term holding both letters is a product of two functions of x, so it yields two pieces of which only one carries dy/dx.
The gradient formula that emerges normally mentions both letters, which is not a defect: a point has to be supplied before a number can appear.
The second cue is a variable sitting up in an exponent, as in an expression raised to the power 2x, or four and more moving factors piled into one fraction. The power rule wants a fixed exponent and the exponential rule wants a fixed base, so neither can touch the first of those.
Take natural logarithms of both sides before differentiating.
The exponent drops in front, products become sums and quotients become differences, the left side turns into dy/dx divided by y, and multiplying back by the original expression at the end returns the gradient.
Differentiate again, then look for the pattern
Differentiating a second time gives the second derivative, which reports how fast the gradient itself is shifting, and a third and fourth are written the same way.
A question asking for a derivative of very high order is never asking you to do that many rounds. Do three or four, line the results up, and state the general term in n, treating the sign for odd and even n separately. Two finishing habits are worth keeping.
When a second derivative comes out of a relation, replace every surviving dy/dx by the first derivative formula you already built, because a gradient left sitting inside a second derivative is a wrong answer rather than an unfinished one. And substitute numbers only at the very end: differentiating first and evaluating last keeps the algebra short and leaves you a clean arithmetic check to run.
What this chapter covers
- 01
The derivative as a limit of difference quotients
- 02
Gradient, rate of change, and the units of dy/dx
- 03
Product, quotient and chain rules, and choosing between them
- 04
Derivatives of exponentials and logarithms
- 05
Implicit differentiation on a relation you cannot untangle
- 06
Logarithmic differentiation and higher order derivatives
Worked example · free
Choosing the rule twice in one question
- 1Name the division as the last operation in part (a), and note that its top is a composite, so the chain rule runs inside the quotient rule.
- 2Apply the quotient rule, keeping the whole top inside one bracket.
- 1Put x = 1 into the derivative and reduce the fraction.
- 2For part (b), differentiate every term where it stands, using the product rule on the mixed term and writing dy/dx wherever y was differentiated.
- 2Collect the dy/dx terms, factor, divide, and only then substitute the point.
Key terms
- Difference Quotient
- A difference quotient is the change in a function's output divided by the change in its input across a step of size h. Its limit as h shrinks to 0 is the derivative at the starting point.
- Chain Rule
- The chain rule differentiates a composite by multiplying the derivative of the outer function, evaluated at the inner one, by the derivative of the inner function. A composite with three layers therefore hands over three factors.
- Implicit Differentiation
- Implicit differentiation treats y as an unnamed function of x and differentiates a relation term by term where it stands. It produces a dy/dx at every place where something containing y was differentiated.
- Logarithmic Differentiation
- Logarithmic differentiation takes the natural logarithm of both sides before differentiating, so that a moving exponent drops in front and a long product turns into a sum. Multiplying back by the original expression recovers the gradient.
- Second Derivative
- The second derivative is the derivative of the first derivative, so it measures how quickly the gradient itself is changing. It is written as f double prime or as d squared y over dx squared.
Differentiation Rules and Techniques FAQ
When is it worth taking logarithms before differentiating?
Take them when the variable sits up in an exponent, so that neither the power rule nor the exponential rule applies, or when four or more moving factors are piled into one fraction. Logarithms turn that exponent into a coefficient and the product into a sum, which the ordinary rules can then handle comfortably.
Why does my answer still contain y after I differentiate a relation?
Because the relation never defined y as a single formula in x, so the gradient genuinely depends on which point of the curve you are standing at. Supply both coordinates and the formula returns one number, and a curve that crosses itself can carry two different gradients at the same value of x.
Exam move
Take twenty expressions off the exercise sheets and differentiate none of them. Beside each one write two things only: the operation performed last, and the rule that follows from it. Then differentiate just the five that made you hesitate, and check each of those numerically by stepping a small amount either side of a convenient input.
What separates a fast attempt from a slow one in this chapter is the rule choice at the start, not the algebra that comes after it.
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