Single variable function
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Why functions
The study of natural phenomena often leads us to consider variations in one physical quantity as a function of variations in another. Thus, the force exerted between two electric charges \(g\) and \(q'\) is related to the distance \(r\) separating them by Coulomb's Law:
If distance varies, force takes on different values. We say that the force is a function of the distance and we note:
More generally, we'll say that \(y\) is a function of the variable \(x\) and we'll write \(y = f(x)\) if, for each value of \(x\), there's a corresponding value of \(y\).The set of values of \(x\) for which \(y = f(x)\) exists is called the "domain" or "interval" of definition of the function.
Derivative of a single variable function
Definition
Let \(y = f(x)\) be a function defined and continuous in the vicinity of the point of abscissa \(x = x_0\). If \(x\) is given an increase \(\Delta x\) (positive or negative) from \(x_0\), \(y\) varies from :
We call \(f'(x_0)\), the limit, if it exists, towards which the ratio \(\frac{\Delta y}{\Delta x}\) tends as \(\Delta x\) approaches zero.
Let's give some examples of direct calculation of derivatives.
Example No. 1: calculating the derivative of \(y = x^2\)
\(f(x) = x^2\)
\(f(x + \Delta x) = (x + \Delta x)^2 = x^2 + 2x \Delta x + (\Delta x)^2\)
\(f(x + \Delta x) - f(x) = 2x \Delta x + (\Delta x)^2\)
then,
\(y' = f'(x) = \lim_{\Delta x \to 0} \frac{f(x + \Delta x) - f(x)}{\Delta x} = \lim_{\Delta x \to 0} (2x + \Delta x) = 2x\)
Example No. 2: calculating the derivative of \(y = \sin x\)
\(f(x) = \sin x\)
\(f(x + \Delta x) = \sin(x + \Delta x)\)
\(f(x + \Delta x) - f(x) = \sin(x + \Delta x) - \sin x = 2 \cos \left( x + \frac{\Delta x}{2} \right) \sin \left( \frac{\Delta x}{2} \right)\)
then,
\(y' = \lim_{\Delta x \to 0} \frac{\cos \left( x + \frac{\Delta x}{2} \right) \sin \left( \frac{\Delta x}{2} \right)}{\Delta x / 2} = \cos x\)
and because the ratio \(\sin \left( \frac{\Delta x}{2} \right) / \left( \frac{\Delta x}{2} \right)\) tends towards 1 as \(\Delta x \to 0\).
This is only true if \(\Delta x\) is expressed in radians. This must be remembered especially when calculating limited developments (see in other section).
Derived functions
If a function \(f(x)\) admits a derivative for all values of \(x\), we define a function \(f'(x)\) which we call the "derived function" and we note:
GEOMETRICAL INTERPRETATION OF THE DERIVATIVE
Consider the graph of the function \(y = f(x)\) and let M and N be the points with abscissas \(x_0\) and \(x_0 + \Delta x\). We see that:
\(\Delta x=MP\) and \(\Delta y = PN\) and \(\frac{\Delta y}{\Delta x} = \tan \beta\)
When \(\Delta x\) tends towards zero, N approaches M and the secant (S) tends towards the tangent (T) to the curve at point M, the angle \(\beta\) simultaneously tends towards \(\alpha\).
The derivative of the function \(f(x)\) at the abscissa \(x = x_0\) is numerically equal to the slope of the tangent to the curve at point \((x_0, y_0)\)
\(f'(x_0) = \tan \alpha\)
Remark: The notation "tg" has been retained throughout the work for the tangent of an angle. This notation is equivalent to "tan".
CALCULATION OF DERIVATIVES
Derivative of a sum, of a product and quotient
\(U(x)\) and \(V(x)\) being both differentiable functions, we demonstrate using the very definition of the derivative that:
\(\([U(x)+V(x)]' = U'(x) + V'(x)\)\)
' = U'(x) cdot V(x) + U(x) cdot V'(x)
' = frac{U'(x) cdot V(x) - U(x) cdot V'(x)}{V^2(x)}
Example No. 1
\(y = 4x^3 + 6x + 2\)
\(y' = 12x^2 + 6\)
Example No. 2
\(y = x^2(x^3 - 1)^2\)
\(y' = 2x(x^3 - 1)^2 + 6x^2(x^3 - 1)\)
Derivative tab
| Function | Derivative |
|---|---|
| \(y = a\) | \(y' = 0\) |
| \(y = ax\) | \(y' = a\) |
| \(y = x^m\) | \(y' = mx^{m-1}\) |
| \(y = U(x)^m\) | \(y' = mU(x)^{m-1}U'(x)\) |
| \(y = \frac{1}{x}\) | \(y' = -\frac{1}{x^2}\) |
| \(y = \frac{1}{U(x)}\) | \(y' = -\frac{U'(x)}{U(x)^2}\) |
| \(y = \sqrt{x}\) | \(y' = \frac{1}{2\sqrt{x}}\) |
| \(y = \sqrt[m]{U(x)}\) | \(y' = \frac{U'(x)}{m\sqrt[m]{U(x)^{m-1}}}\) |
| \(y = \sin x\) | \(y' = \cos x\) |
| \(y = \sin U(x)\) | \(y' = U'(x) \cos U(x)\) |
| \(y = \cos x\) | \(y' = -\sin x\) |
| \(y = \cos U(x)\) | \(y' = -U'(x) \sin U(x)\) |
| \(y = \tan x\) | \(y' = 1 + \tan^2 x = \frac{1}{\cos^2 x}\) |
| \(y = \tan U(x)\) | \(y' = U'(x)(1 + \tan^2 U(x)) = U'(x)\frac{1}{\cos^2 U(x)}\) |
| \(y = \cot x\) | \(y' = -1 - \cot^2 x\) |
| \(y = \cot U(x)\) | \(y' = -U'(x)(1 + \cot^2 U(x)) = -U'(x)\frac{1}{\sin^2 U(x)}\) |
| \(y = a^x\) | \(y' = a^x \ln a\) |
| \(y = a^{U(x)}\) | \(y' = a^{U(x)}U'(x) \ln a\) |
| \(y = e^x\) | \(y' = e^x\) |
| \(y = e^{U(x)}\) | \(y' = e^{U(x)}U'(x)\) |
| \(y = \ln x\) | \(y' = \frac{1}{x}\) |
| \(y = \ln U(x)\) | \(y' = \frac{U'(x)}{U(x)}\) (dérivée logarithmique) |
| \(y = \log x\) | \(y' = \frac{1}{x \ln 10}\) |
| \(y = \log U(x)\) | \(y' = \frac{U'(x)}{U(x)\ln 10}\) |
| \(y = \sinh x\) | \(y' = \cosh x\) |
| \(y = \sinh U(x)\) | \(y' = U'(x)\cosh U(x)\) |
| \(y = \cosh x\) | \(y' = \sinh x\) |
| \(y = \cosh U(x)\) | \(y' = U'(x)\sinh U(x)\) |
| \(y = \tanh x\) | \(y' = \frac{1}{\cosh^2 x}\) |
| \(y = \tanh U(x)\) | \(y' = \frac{U'(x)}{\cosh^2 U(x)}\) |
| \(y = \mathrm{Arcsin} x\) | \(y' = \frac{1}{\sqrt{1 - x^2}}\) |
| \(y = \mathrm{Arccos} x\) | \(y' = -\frac{1}{\sqrt{1 - x^2}}\) |
| \(y = \mathrm{Arctan} x\) | \(y' = \frac{1}{1 + x^2}\) |
| \(y = \mathrm{Arcsin} U(x)\) | \(y' = \frac{U'(x)}{\sqrt{1 - U(x)^2}}\) |
| \(y = \mathrm{Arccos} U(x)\) | \(y' = -\frac{U'(x)}{\sqrt{1 - U(x)^2}}\) |
| \(y = \mathrm{Arctan} U(x)\) | \(y' = \frac{U'(x)}{1 + U(x)^2}\) |
Composition
inspi : https://medium.com/y-math/math-memes-post-1-memeifying-math-fa77985a4bc1