**Derivative of logarithmic functions**

Derivative of log

Derivative of log

_{b}xd/dx (log

_{b}x) = 1/xlnb

To remember this formula let's apply the following technique:

1) Multiply the number of which we calculate the logarithm by the natural logarithm of the base. The number here is x and the base is b. Therefore we have xlnb

2) Take the inverse of this product. The inverse of the product is 1/xlnx

**Derivative of lnx**

d/dx(lnx) = 1/x

The derivative of the logarithm of any number is equal to the inverse of this number.

**Derivative of log**

_{b}uSince log

_{b}u is a composite function its derivative is given by d/dx(log

_{b}u) = d/du(log

_{b}u).du/dx

d/dx(log

_{b}u) = 1/ulnnb.du/dx

**Rule: The derivative of the logarithm of a composite function is equal to its derivative with respect to the new variable (u) multiplied by the derivative of the new variable (u) with respect to x.**

**Derivative of lnu**

**Since u**

**is a composite function we have d/dx(lnu) = d/du(lnu).du/dx**

= i/u.du/dx

**Rule: The derivative of the natural logarithm of a composite function u is equal to the inverse of the function multiplied by its derivative with respect to x**

**Example 1, Calculate the derivative of y = x³log**

_{5}2x

The derivative of y is y" = (x³log

_{5}2x)'

Let's apply the product rule:

Y' = (x³)'(log

_{5}2x) + x³(log

_{5}2x)'

The derivative of x³ is obvious. Let's calculate the derivative of log

_{5}2x

Let's write u = 2x we have (log

_{5}u)' = d/du(log

_{5}u),du/dx

= i/uln5.u'

= 1/2x.ln5.(2x)'

= 1/2x.ln5.2

= 1/xln5

let's go back to the derivative of y we have:

y' = 3x²log

_{5}2x + x³.1/xlnx

= 3x²log

_{5}2x+x²/lnx

Example 2. Calculate the derivative of y = ln(2x²-4x+3)

Let's write u = 2x²-4x+3

We have y = lnu

Then dy/dx = d/dx(lnu)

Since lnu is a composite function then dy/dx = d/du(lnu).du/dx

= 1/u(4x-4)

Substitute u: dy/dx = (1/2x³-4x+3).(4x-4)

dy/dx = 4x-4/2x²-4x+3

= 4(x-1)/2x³-4x+3

**Practice**

**Calculate the derivative of the following functions;**

1.log₅(2x+5)

2. 5/log(x+4)

3. ln(sinx)

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