a^{x}

a^{x} ∙ lna, where a is a constant

e^{x}

e^{x}

e^{c∙}^{x}

c∙e^{c∙x}, where c is a constant

x^{x}

x^{x}(1+ln(x))

log(x), where the base is 10

1/(x∙ln(10))

log_{a(}x), where the base is a

1/(x∙ln(a))

ln(x)

1/x

f^{ g}, f and g are both functions

f^{ g}(g '∙ln(f)+(g/f)∙f ')

Example 1: Find the derivative of f(x) = ln(tan x).
f´(x) = 1 / tan(x) * d/dx * tan(x)
f´(x) = 1 / tan(x) * sec^{2}(x)
Example 2: Find the derivative of f(x) = e^{(2x1)}
f´(x) = e^{(2x1)} * d(2x 1 ) / dx
f´(x) = e^{(2x1)} * 2
Example 3: Find d(3^{x}) / dx
d(3^{x}) / dx = 3^{x}ln3
Example 4: Find the derivative of x^{x2}
Let y =x^{x2}
Take natural logarithm on both the sides
lny = (x2)lnx
We now differentiate both sides with respect to x, using chain rule on the left side and the product rule on the right.
Multiply y on the right hand side
dy/dx = y [lnx*1 + (x2)/x]
dy/dx = x^{x2}[lnx*1 + (x2)/x]
dy/dx =x^{x3}(x*lnx + x – 2)