Differentiability

What Nour says

1. 1. Differentiability at a point, derivative number

   Today, the whole chapter: "Differentiability", Year 4, Mathematics section.

   We will build this step by step: what differentiability means, how it shapes a curve, and two key theorems.

   Each part has its own worked example, close to what you see in the bac.

   Let's start with section one: differentiability at a point, and the derivative number.

   Take a function f and a point a inside its domain.

   We look at the ratio between the change in f and the change in x.

   This ratio is called the rate of change between a and x.

   f is differentiable at a if this ratio has a finite limit when x tends to a.

   That limit, when it exists, is called the derivative number of f at a.

   In other words, f prime of a measures how fast f changes right at a.

   We can also write this limit with h instead of x minus a.

   Both forms say exactly the same thing, just with a different letter.

   Now let's see what this means on a curve.

   Here is the curve of f, and a point A on it, at x equals a.

   Take a second point M on the curve, close to A.

   The rate of change is the slope of the line through A and M.

   As M slides toward A, this line turns and gets closer to one fixed position.

   That fixed position is called the tangent line to the curve at A.

   Its slope is exactly the derivative number f prime of a.

   This is why the derivative number matters: it gives the direction a curve follows.

   When f is not differentiable at a, the curve has no clear tangent there.

   A classic case is the absolute value function at x equals 0.

   From the right, the rate of change tends to 1, from the left it tends to minus 1.

   The two one-sided limits differ, so there is no single limit, and no derivative at 0.

   On the curve, this shows up as a sharp corner, not a smooth tangent.

   Now, let's work through a full example, step by step.

   Let f of x equal x squared plus 1, and study differentiability at a equals 2.

   We form the rate of change between x and 2.

   f of 2 equals 5, since 2 squared plus 1 is 5.

   We simplify the numerator: x squared minus 4, which factors as x minus 2 times x plus 2.

   We cancel x minus 2, since x is not equal to 2.

   Now we let x tend to 2 in this simplified expression.

   The limit is finite, so f is differentiable at 2, and f prime of 2 equals 4.

   We can picture this on the curve of f, which is a parabola.

   At the point where x equals 2, f of 2 equals 5.

   The tangent there has slope 4, matching our computed value f prime of 2.

   So the derivative number is both a limit of ratios and a slope on the graph.

   To sum up: differentiability at a point means this special limit exists and is finite.

   The derivative number f prime of a is the value of that limit.

   Geometrically, it gives the slope of the tangent line at that point on the curve.

   What is the rate of change used to define f'(a), as a limit?

2. 2. Differentiable function, derivative function

   Let's move to section two: differentiable functions and the derivative function.

   A function f is differentiable on an interval if it is differentiable at every point of it.

   Then at each point x of that interval, we get one derivative number f prime of x.

   This new function, built from all those derivative numbers, is called the derivative function.

   So f gives a curve, and f prime gives a new function, built point by point.

   We already know some derivative functions from earlier work.

   Also the derivative of a constant is 0, and the derivative of x is 1.

   For sums, products and quotients, fixed rules let us compute f prime without limits each time.

   Here is why this matters: f prime tells us where f increases or decreases.

   When f prime is positive, the curve of f climbs upward.

   When f prime is negative, the curve of f goes downward instead.

   And where f prime equals 0, the curve has a flat point, a peak or a valley.

   Now, a full worked example, close to a bac exercise.

   Let f of x equal x cubed minus 3x, on the real numbers.

   f is a polynomial, so it is differentiable everywhere, on all of the reals.

   We apply the rule for x cubed and the rule for minus 3x.

   We factor 3 out, giving 3 times x squared minus 1.

   This is 0 when x equals 1 or x equals minus 1.

   Between minus 1 and 1, the factors give f prime negative, so f decreases there.

   Outside that interval, f prime is positive, so f increases on both sides.

   This gives us the full shape of the curve, just from studying f prime's sign.

   To sum up: on an interval, f differentiable gives a new function f prime.

   The sign of f prime tells us exactly where f increases and where it decreases.

   This sign study is the main tool for understanding any curve's shape.

   For f(x) = x³ − 3x, what is f'(x)?

3. 3. Second derivative and point of inflection

   Section three: the second derivative, and the point of inflection.

   If the derivative function f prime is itself differentiable, we can derive it again.

   This new function is called the second derivative, written f double prime.

   That is, the second derivative is the derivative of the derivative, measuring how f prime itself changes.

   f prime's sign tells us if f increases or decreases.

   In the same way, f double prime's sign tells us about the curve's bending.

   When f double prime is positive, the curve bends upward, like a cup.

   We say the curve is convex on that part.

   When f double prime is negative, the curve bends downward, like a cap, and is concave.

   A point of inflection is where the curve switches from convex to concave, or back.

   That is, the point where the bending direction of the curve changes, like an S-shape.

   At such a point, f double prime equals 0, and changes sign around it.

   Beware: f double prime equals 0 alone is not enough, the sign must truly change.

   Now let's work a full example, similar to a bac question.

   Let f of x equal x cubed minus 3x squared plus 2, on the reals.

   First we find f prime, using the power rule on each term.

   Now we derive again to get f double prime.

   We factor, giving 6 times x minus 1.

   This is 0 only at x equals 1.

   For x less than 1, x minus 1 is negative, so f double prime is negative.

   For x greater than 1, f double prime becomes positive, so the curve turns convex.

   The sign truly changes at x equals 1, so this is a point of inflection.

   We find its y-coordinate by computing f of 1.

   So the inflection point has coordinates 1 and 0.

   Before x equals 1, the curve is concave, after, it becomes convex, exactly as we found.

   To sum up: the second derivative measures how the curve bends, convex or concave.

   A point of inflection is where that bending direction actually switches.

   What condition must f'' satisfy at a point of inflection?

4. 4. Mean value theorem

   Section four: the Mean Value Theorem.

   This theorem is also called the theorem of finite increments.

   Suppose f is continuous on the closed interval a, b, and differentiable on the open interval a, b.

   Then there exists at least one point c strictly between a and b with a special property.

   At that point c, the derivative equals the rate of change between a and b.

   That is, somewhere between a and b, the tangent slope matches the overall average slope.

   Let's see this on a curve to understand it fully.

   Here is the curve of f, between the points A and B.

   The straight line through A and B is called the secant line.

   Its slope is exactly the rate of change between a and b.

   The theorem says some point C on the curve has a tangent parallel to this secant.

   This makes sense: somewhere, the curve must run parallel to the straight path from A to B.

   Now let's apply this to a concrete worked example.

   Let f of x equal x squared, on the interval 1 to 3.

   f is a polynomial, so it is continuous and differentiable everywhere, hence on this interval too.

   The theorem guarantees some c between 1 and 3 with f prime of c equal to this rate.

   f of 3 is 9, and f of 1 is 1, so the numerator is 8.

   So we need f prime of c equal to 4.

   Since f prime of x equals 2x, we solve 2c equals 4.

   This gives c equal to 2, which does lie between 1 and 3.

   So the theorem is verified, with this exact value of c.

   To sum up: the theorem links the average rate of change to one tangent slope.

   It guarantees a point c exists, but it does not always tell us how to find it easily.

   In the mean value theorem, what equation must f'(c) satisfy?

5. 5. The inequality of finite increments

   Section five: the inequality of finite increments.

   This inequality is a very useful consequence of the mean value theorem.

   Suppose f is differentiable on an interval, and f prime stays between two bounds m and M there.

   Then for any a and b in that interval, f of b minus f of a is controlled too.

   That is, bounding the slope of f prime bounds how much f itself can change.

   A common special case uses just one bound, written with absolute value.

   Here is why this holds: the mean value theorem gives some c with f prime of c equal to the rate of change.

   Since f prime of c sits between m and M, the whole ratio does too.

   Multiplying through by b minus a gives exactly the inequality we stated.

   This tool is strong: it bounds f without ever solving for the exact value of c.

   Now, a full worked example, in a bac style.

   Let f of x equal the square root of x, on the interval 4 to 9.

   f prime of x equals 1 over 2 square root of x.

   On this interval, x is between 4 and 9, so square root of x is between 2 and 3.

   Taking reciprocals and halving, f prime of x lies between 1 over 6 and 1 over 4.

   Now we apply the inequality with a equal 4 and b equal 9, so b minus a equals 5.

   This gives f of 9 minus f of 4 between 5 over 6 and 5 over 4.

   We found this bound without ever computing the exact value of c.

   To sum up: this inequality bounds the change in f using bounds on f prime.

   It is a direct and powerful use of the mean value theorem.

   What does the inequality of finite increments need as its hypothesis on f'?

6. 6. Derivative of a composite function, affine approximation

   Section six, our last: the derivative of a composite function, and affine approximation.

   A composite function is built by applying one function, then another, to the result.

   That is, we first compute u of x, then apply f to that result.

   There is a direct rule to derive this composite function.

   We first derive the inner function, then multiply by the derivative of the outer one.

   A frequent special case is u of x raised to the power n.

   Another frequent case is the square root of u of x.

   Here is why the chain rule works: a small change in x shifts u, then shifts f.

   The two small changes multiply together, which gives the product in the formula.

   Now let's see affine approximation, our last tool in this chapter.

   That is, near a point a, we replace the curve of f by its tangent line.

   For x close to a, f of x is close to this simple linear expression.

   On the graph, this line is just the tangent at the point a.

   Far from a, the tangent drifts from the curve, so this approximation stays only local.

   Let's work through one full bac-style example.

   Let f of x equal square root of x, near a equals 4.

   f of 4 equals 2, since 4 is a perfect square.

   f prime of x equals 1 over 2 square root of x, so f prime of 4 is 1 over 4.

   We plug these into the affine approximation formula near a equals 4.

   We use this to estimate the square root of 4.1, without a calculator.

   This gives an approximate value close to 2.025.

   This matches the real value of square root of 4.1 very closely.

   To sum up: the chain rule derives functions built from other functions, one inside another.

   Affine approximation replaces a curve by its tangent, giving fast estimates near one point.

   In the affine approximation formula, what does f'(a) multiply?

   That completes the chapter on differentiability, from the derivative number to affine approximation.

   Keep these tools close: sign of f' for variations, sign of f'' for shape, and the mean value theorem linking them.

   Practice both exercises, and you will be ready for this part of the bac.