Math Help

Derivatives of Trig, Exponential, Log functions + Logarithmic Differentiation

In a Nut Shell: You must know the derivatives of basic functions appearing in calculus.

Basic functions include trigonometric functions, exponential functions and logarithmic

functions. For combinations of these functions to form more complicated ones you

should probably use the “chain rule” and/or logarithmic differentiation.

Derivatives of Trigonometric Functions are: (You must know these derivatives.)

d/dx[sin x ] = cos x, d/dx[cos x] = - sin x, d/dx[tan x] = sec² x,

d/dx[cot x ] = -csc²x, d/dx[sec x] = sec x tan x, d/dx[csc x] = -csc x cot x

The derivative of the exponential function e ^x is as follows:

d/dx[ e ^x] = e ^x

The derivative of the exponential function e ^ax is as follows:

d/dx[ e ^ax] = a e ^ax

Logarithms can be to any base, a. i.e. y(x) = log_a(x)

If the base happens to be e then log_e(x) = ln(x) which is the natural logarithm.

The derivative of the natural logarithm, which means the logarithm to the base e is:

y = ln x, then dy/dx = d/dx[ln x] = 1/x

Suppose u = u(x) and y = ln u . Now apply the chain rule to calculate d/dx [ ln u ]

The chain rule for this application is dy/dx = (dy/du) (du/dx)

dy/du = 1/u so dy/dx = (1/u) du/dx

For derivatives of a logarithmic function to any base, a, other than e you must first convert

to base e prior to calculating the derivative.

The conversion from any base a to base e is as follows: log_a(x) = ln(x) / ln(a).