Differential calculus

The equation for the derivative of a function is:

f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}

The derivative of a function is only defined where the function is defined and is continuous (not a break, gap, sharp point, or end).

Notation for differentiation

Leibniz notation for the first derivative is \frac{\dy}{\dx}, or \frac{\dd}{\dx}[y], or \frac{\d{f(x)}}{\dx}, or \frac{\dd}{\dx}[f(x)]. Notation for the second derivative is \frac{d^2y}{\d{x^2}}. \dy represents an infinitesimal change in y, whereas \Delta y would represent a real change in y.

Lagrange notation for the first derivative of f(x) is f'(x), and for the second derivative is f''(x).

Basic rules for differentiation

First, every term in the expression that does not contain a bound variable is removed. These terms are all constants and do not contribute to the gradient of the function.

\frac{\dd}{\dx}[5 + 2a] = 0

For every term in the expression, we multiply the term by the power and then decrement the power by 1:

\frac{\dd}{\dx}[x^n] = nx^{n-1}

The derivative of the sum is the sum of the derivatives:

f(x) = g(x) + h(x) \\ f'(x) = g'(x) + h'(x)

Chain rule

To find the derivative of a function g(x) to the power of n, multiply by the power and reduce the power by 1 as usual, but also multiply by the derivative g'(x) of the function. In other words, n \times g'(x) \times g(x)^{n-1}.

\frac{\dd}{\dx}([g(x)]^n) = n \times g'(x) \times [g(x)]^{n-1}, n \ne 0

To find the derivative of a trigonometric function applied to a function g(x), multiply by the derivative g'(x) of the inner function, and replace the trigonometric function with its derivative.

\frac{\dd}{\dx}[sin(g(x))] = g'(x) \times cos(g(x))

To find the derivative of any composite function f(g(x)):

\frac{\dd}{\dx}[f(g(x))] = f'(g(x)) \times g'(x)

A second, equivalent formula is given as follows:

\frac{\d{f}}{\dx} = \frac{\d{f}}{\d{u}} \times \frac{\d{u}}{\d{x}}

As an example, the equation h(x) = (3x+1)^{10} has h(u) = u^{10} and u(x) = 3x+1.

Product rule

To find the derivative of a product of two functions f(x) and g(x):

\frac{\dd}{\dx}[f(x)g(x)] = f'(x)g(x) + g'(x)f(x)
(fg)' = f'g + g'f

Quotient rule

\frac{\dd}{\dx}\left[\frac{f(x)}{g(x)}\right] = \frac{f'(x)g(x) - f(x)g'(x)}{(g(x))^2}
\frac{f}{g}' = \frac{f'g - fg'}{g^2}

Concavity

If the slope of a function is increasing (the second derivative of the function at that point is positive), the function at that point is concave up (looks like a cup), and a turning point is a relative minimum.

If the slope of a function is decreasing (the second derivative of the function at that point is negative), the function at that point is concave down (looks like an umbrella), and a turning point is a relative maximum.

Critical points

A critical point is a point where the derivative is zero or undefined.

To find the critical points of a function, first find the derivative of the function. Check for all values of x where the derivative is undefined, and then solve the derivative for zero. Finally, substitute the solved x values into the original equation to find the full coordinates of every critical point, if that point isn’t an asymptote.

Turning points

A turning point is a point where the slope of a function changes from negative to positive (or vice versa).

The first derivative of the function at a turning point is zero. If the slope changes from positive to negative (the second derivative at that point is negative), the point is a relative maximum. If the slope changes from negative to positive (the second derivative at that point is positive), the point is a relative minimum. Otherwise, the point is a point of inflection, not a minima or maxima.

Point of inflection

A point of inflection is a point where the concavity of the function changes. The first derivative at a point of inflection is not necessarily zero.

If the second derivative of a function at a point is zero (the second derivative changes sign at this point), the point is a point of inflection.

Second derivative test

To determine whether a point (where the first derivative is zero) is a minimum or a maximum, we can check the sign of the second derivative at that point:

  • if the second derivative is negative, the slope is decreasing, and the point is a relative maximum
  • if the second derivative is positive, the slope is increasing, and the point is a relative minimum
  • if the second derivative changes sign (passes through zero), the point is a point of inflection

First derivative test

This test is for cases where the second derivative is tedious to find.

If the first derivative at a point is zero and:

  • changes sign from positive to negative, the point is a relative maximum (concave down)
  • changes sign from negative to positive, the point is a relative minimum (concave up)
  • does not change sign, it is neither a minimum nor a maximum, but could be a point of inflection

To check how the sign changes, we calculate the value of points to the left and right.