Derivative of a function : 11th grade math lesson to download in PDF

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The derivative of a function in a 11th grade math lesson where we will find the derivative at a point and the concrete meaning of the derivative number and the equation of the tangent at a point.
In this eleventh grade lesson, we will discuss the derivative of a sum, product and quotient. The derivative and the direction of variation of a function, as well as the derivatives of usual functions.

I. Derivative number – derivative function – tangent to a curve.

Definition:

Let f be a function defined on an interval I. The curve (C) below is the graphical representation of f in an orthonormal frame of reference (O,\vec{i},\vec{j}). M and N are two points of (C) with respective abscissas a\in\,Iand x=a+h\in\,Iwhere h\in\,\mathbb{R}^*.

Derivative and rate of increase

M and N therefore have the following coordinates: M(a;f(a))and N(x;f(x)), i.e.: N(a+h;f(a+h)).

So we have: \vec{MN}\,(\,x-a;f(x)-f(a)\,) or \vec{MN}\,(\,h;f(a+h)-f(a)\,)

The directrix of the line (MN) secant to (C) is therefore

m=\frac{f(x)-f(a)}{x-a}=\frac{f(a+h)-f(a)}{h}.

If the curve (C) has at M a tangent of directing coefficient d, then when the point N approaches M, that is to say when x tends to a, or, which is the same, when h tends to 0,

the secants (MN) will reach a limit position which is that of the tangent(MP) in M to (C).

This can then be translated using the governing coefficients as:

\lim_{x\to\,a}\frac{f(x)-f(a)}{x-a}=d i.e.: \lim_{h\to\,0}\frac{f(a+h)-f(a)}{h}=d.

So we have: \lim_{h\to\,0}\,%5B\frac{f(a+h)-f(a)}{h}-d\,\,%5D=0.

If we call \Phi, the function defined for h\in\,\mathbb{R}^* and a+h\in\,I by :

\Phi\,(h)=\frac{f(a+h)-f(a)}{h}-d.

we have :

\lim_{h\to\,0}\Phi\,(h)=0 and h\Phi\,(h)=\,f(a+h)-f(a)-dh, which is also written: f(a+h)=\,f(a)+dh+h\Phi\,(h).

Conversely, if there exists a real d and a function \Phi such that, for all h\in\,\mathbb{R}^* and a+h\in\,I, we have: f(a+h)=\,f(a)+dh+h\Phi\,(h) with \lim_{h\to\,0}\Phi\,(h)=0,

we deduce that: \Phi\,(h)=\frac{f(a+h)-f(a)}{h}-d and therefore that: \lim_{h\to\,0}\frac{f(a+h)-f(a)}{h}=d.

This allows us to give the three equivalent definitions:

Definition 1:

If f is a function defined on an interval I and if a\in\,I.

When there is a real number d such that, for any real h close to 0, we have

\lim_{h\to\,0}\frac{f(a+h)-f(a)}{h}=d

We say that the function f is derivable in a and that d=f'(a) is the derivative of f in a.

Definition 2:

If f is a function defined on an interval I and if a\in\,I.

When there is a real number d such that, for any real x\in\,Iand close to a, we have:

\lim_{x\to\,a}\frac{f(x)-f(a)}{x-a}=d

We say that the function f is derivable in a and that d=f'(a) is the derivative of f in a.

II. Derivative function on an interval I. Derivative function of a function derivable on I

Definition:

We say that f is derivable on an interval I when it is derivable at any point of I.

When f is derivable on an interval I, the function which associates to any x\in\,I the derivative number of f at x is called the derivative function of f on I. This function is noted f'.

Graphical interpretation of the derivative number.

If f is a function defined on an interval I. If a\in\,Iand if f is derivable at x=a, then :The representative curve of f has a tangent at the point M(a;f(a)) and the directing coefficient of this tangent is the derivative number f'(a)of the function f at x=a.

Remarks:

If the graph of f does not have a tangent at the point M with abscissa x=a, then the function f is not derivable at a. This is the case for the absolute value function in x=0.

The graph of a function may well have a tangent at a point without the function being derivable at that point: it is sufficient that the directing coefficient of this tangent does not exist (tangent parallel to the ordinate axis).

This is the case for the square root function in x=0.

III. Equation of the tangent to a curve

Definition:

If function f is derivable at a, the tangent (MP) to the curve (C) at M of abscissa x=a exists. It has the directing coefficient m=f'(a).

Its equation is therefore of the form: y=mx+p, where m=f'(a) and its intercept p can be calculated.

It is sufficient to write that (MP) passes through m=f'(a).

So we have: f(a)=f'(a)\times  \,a+p. This gives: p=f(a)-a\times  \,f'(a).

So: y=f'(a)x+f(a)-af'(a)which is often written in one of the forms, easier to remember:

Equation of the tangent at the point M(a;f(a)):

Definition:

y=f'(a)(x-a)+f(a) or y-f(a)=f'(a)(x-a).

IV. Sign of the derivative and direction of variation of a function

We admit without proof the following theorems:

Theorem 1:

Let f be a differentiable function on an interval I.

  • If f is increasing on I, then for all x\in\,I, we have f'(x)\geq\,\,0
  • If f is decreasing on I, then for all x\in\,I, we have: f'(x)\leq\,\,0.
  • If f is constant on I, then for all x\in\,I, we have: f'(x)=\,0.
Theorem 2:

Let f be a differentiable function on an interval I.

  • If, for all x\in\,I, we have: f'(x)\geq\,\,0, then f is increasing on I.
  • If, for all x\in\,I, we have: f'(x)\leq\,\,0, then f is decreasing on I.
  • If, for all x\in\,I, we have: f'(x)=\,0, then f is constant on I.
Theorem 3:

Let f be a differentiable function on an interval I.

  • If, for all x\in\,I, we have: f'(x)>\,0 ( except perhaps at isolated points where f'(x)=\,0), then f is strictly increasing on I.
  • If, for all x\in\,I, we have: f'(x)<\,0 ( except perhaps at isolated points where f'(x)=\,0), then f is strictly decreasing on I.

In particular:

Let f be a function derivable on an interval %5Ba;b%5D.

Ownership:
  • If, for any x\in%5Ba;b%5D, we have f'(x)>\,0, then f is strictly increasing on %5Ba;b%5D.
  • If, for any x\in%5Ba;b%5D, we have f'(x)<\,0, then f is strictly decreasing on %5Ba;b%5D.

Examples:

1) Let the function f be defined on \mathbb{R} by f(x)=x^2.

f is derivable on \mathbb{R} and f'(x)=2x for all x\in\,\mathbb{R}.

– For all x\in\,%5D-\infty;0%5D, we have f'(x)\leq\,\,0, so f is decreasing on %5D-\infty;0%5D.

– For allx\in\,%5B0;+\infty%5B, we have f'(x)\,\geq\,\,0, so f is increasing on \,%5B0;+\infty%5B.

Although f'(0)\,=\,0, we have more precise:

– For all x\in\,%5D-\infty;0%5B, we have f'(x)\,<\,0, so f is strictly decreasing on %5D-\infty;0%5D.

– For all x\in\,%5D0;+\infty%5B, we have f'(x)\,>\,0, so f is strictly increasing on %5D0;+\infty%5B.

V. Change of sign of the derivative and extremum of a function

We admit without proof the following theorems:

Theorem:

If f is a differentiable function on an interval I,

And if f admits a local maximum or a local minimum at x=a different from the extremities of the interval I,

So: f'(a)=0.

1. special case where f is derivable on an open interval.

Ownership:

If f is a differentiable function on an open interval I,

And if f admits a local maximum or a local minimum in a\in\,I,

So: f'(a)=0.

Ownership:

If f is a differentiable function on an open interval I,

and if a\in\,I and if f'(x) cancels for x=a by changing sign,

Then f(a) is a local extremum of f on I.

Examples:

1) Let the function f be defined on \mathbb{R}by f(x)=2x^3-3x^2-12x+5. f is derivable on \mathbb{R} with f'(x)\,=6x^2-6x-12=6(x+1)(x-2).

f'(x) cancels out at x=-1 and x=2 by changing sign, because :

  • for x belonging to -%5D\infty;-1%5B, we have: f'(x)>0 so f is strictly increasing on -%5D\infty;-1%5B
  • .
  • for x belonging to %5D-1;2%5B, we have : f'(x)<0 so f is strictly decreasing on%5D-1;2%5B.
  • for x belonging to %5D2;+\infty%5B, we have : f'(x)>0 so f is strictly increasing on %5D2;+\infty%5B.

The function f thus has a local maximum at x=-1 and a local minimum at x=2.

All this study can be summarized in the table below:

Table of variation of a function

Here is a piece of the graphical representations of f and f':

Curve of the function and its derivative.

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