Math Help

Examples of Derivatives of Trig, Exponential, Log functions

Example: Derivatives of more complicated Trigonometric Functions:

d/dx[sin ax ] = a cos ax, d/dx[cos ax] = - a sin ax

d/dx[tan ax ] = a sec² ax, d/dx[sec ax] = a sec ax tan ax

Example: Find the derivative of y = sin (e^tan^x ) Apply the “chain rule”.

To simplify, let u = tan x and w = e^u . In so doing y = y( w(u(x)) )

Then y = sin w so the chain rule becomes

dy/dx = (dy/dw)(dw/du)( du/dx)

Here dy/dw = cos w, dw/du = e^u and du/dx = sec² x

So dy/dx = cos (w) e^u sec² x = cos (e^tanx) sec² x

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:

d/dx[ln x] = 1/x also if u = u(x) then d/dx[ ln u] = (1/u) du/dx (chain rule)

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 using log_a(x) = ln(x) / ln(a).

Example: y = log₈(x²) Find dy/dx.

Conversion: y = ln(x²) / ln 8 Next apply the chain rule letting u = x²

So y = ln u / ln 8 and dy/dx = (dy/du)(du/dx)

where dy/du = 1 / u ln 8 = 1 / (8 x²) and du/dx = 2x

So dy/dx = 1 / (ln 8 x²) [ 2x] = 2x / ( x² ln 8)