MATH1202 Chap.7 Differentiation, Tangents and Normals
Differentiation, Tangents and Normals
This chapter develops differentiation as a local rate and tangent gradient. It applies power, chain, product and quotient rules with the structure named before algebra begins. Derivative values then construct tangent and normal lines at curve points. Units and graph direction are used as independent checks on contextual rates and calculated signs.
The derivative rule and its value at a particular input are kept distinct: one describes a gradient function, while the other is a number with contextual units. Unsimplified structure is preserved until the correct rule has been applied, reducing lost factors and reversed quotient terms.
What this chapter covers
- 01
Derivative as local rate
- 02
Power and constant rules
- 03
Chain rule structure
- 04
Product and quotient rules
- 05
Tangent line equations
- 06
Normal gradient and special cases
Worked example · free
Differentiate a composite product
- +1Name both factors and differentiate each, including the chain factor.
- +1Apply the product rule without premature expansion.
- +1Evaluate the point and derivative at the requested input.
- +1Use point-gradient form and verify the point lies on the line.
Key terms
- Derivative
- A function measuring local output change per input change.
- Chain Rule
- The derivative rule multiplying an outer derivative by the inner derivative.
- Tangent
- A line matching a curve's local gradient at a point.
- Normal
- A line perpendicular to the tangent at the same curve point.
Differentiation, Tangents and Normals FAQ
How do I recognise the chain rule?
Look for one function used as the input of another, such as a power, exponential or logarithm containing a non-trivial expression. Differentiate the outside while keeping the inside, then multiply by the inside derivative.
Why is the product rule not the product of derivatives?
Both factors change, so the total local change has two contributions: first-factor change times the second factor, plus the first factor times second-factor change. Omitting either contribution loses part of the rate.
How do I check a tangent equation?
Confirm that its gradient equals the derivative value and that the specified curve point satisfies the line equation. These direction and position checks are independent and expose different mistakes.
What is the difference between a derivative and a derivative value?
The derivative is a function that gives the instantaneous rate or tangent gradient at any permitted input. Substituting a particular input produces a derivative value for one point. Keep the variable while finding the general rule, then substitute. Reversing that order can hide structure and prevents the result from answering questions at other inputs.
How do I keep the quotient rule in the right order?
Name the numerator u and denominator v, then write v times u prime minus u times v prime over v squared before substituting. The subtraction order matters. Check the denominator restriction from the original function and simplify only after the full pattern is present. A quick expansion can then expose common factors without changing the rule.
Why is a normal gradient a negative reciprocal?
At a point where the tangent gradient is finite and non-zero, perpendicular line gradients multiply to negative one. Therefore the normal gradient is negative one divided by the tangent gradient. Treat horizontal and vertical tangents separately, because zero has no reciprocal and an undefined tangent gradient corresponds to a horizontal normal.
How do units change under differentiation?
A derivative has output units divided by input units. If distance in metres depends on time in seconds, its first derivative is metres per second and its second derivative is metres per second squared. Writing units beside the result helps distinguish a value from a rate and can expose differentiating the wrong quantity.
What quick checks can I use after differentiating?
Compare the derivative sign with whether the graph is rising or falling, inspect whether a constant disappeared, and verify the units as output per input. For a polynomial, compare the expected degree. For a composite expression, confirm that every inner derivative appears. You can also estimate a nearby secant slope numerically; it should be close to the derivative value when the interval is small.
Exam move
Classify the structure before differentiating. Mark inner functions, name product factors and preserve quotient order. Create mixed drills where the rule name is hidden, then justify the chosen rule before calculating. Keep derivatives factored until the next task requires expansion, and compare a few derivative values with secant slopes or graph direction.
For tangents and normals, calculate the curve point separately from the gradient, then use point-gradient form. Verify the final line with both a gradient check and point substitution. In applications, write the units of the original variables before differentiating and derive the rate units from them. Track recurring errors by rule, especially omitted chain factors, incomplete product terms and reversed quotient subtraction.
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