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# Derivatives of Exponential and Logarithmic Functions

 ax ax ∙ lna, where a is a constant ex ex ec∙x c∙ec∙x, where c is a constant xx xx(1+ln(x)) log(x), where the base is 10 1/(x∙ln(10)) loga(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) * sec2(x)

Example 2: Find the derivative of f(x) = e(2x-1)

f´(x) = e(2x-1) * d(2x -1 ) / dx

f´(x) = e(2x-1) * 2

Example 3: Find d(3x) / dx

d(3x) / dx = 3xln3

Example 4: Find the derivative of xx-2

Let y =xx-2

Take natural logarithm on both the sides

lny = (x-2)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 + (x-2)/x]

dy/dx = xx-2[lnx*1 + (x-2)/x]

dy/dx =xx-3(x*lnx + x – 2)

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