Derivative of ln2x

The derivative of ln2x is equal to 1/x. To calculate this derivative, we can use the chain rule of differentiation or we can evaluate the derivative of ln2x using the logarithmic properties. Differentiation is the process of evaluating the derivative of a function which gives the rate of change of the function with respect to a small change in the variable. The derivative of ln2x is mathematically written as d[ln(2x)] / dx = 1/x.

Let us evaluate the derivative of ln2x and derive its formula using different methods. We will prove the derivative of ln^2x and derive its formula along with a few solved examples involving derivatives of logarithmic function for a better understanding of the concept.

The derivative of ln2x is given by, d[ln(2x)] / dx = 1/x. In general, we can say that the derivative of ln(kx), where k is a real number, is equal to 1/x which can be proved using the chain rule method of differentiation. We can also calculate the derivative of ln(2x) using the logarithmic property given by, log(ab) = log a + log b. Let us explore the formula for the derivative of ln2x in the next section.

The formula for the derivative of ln2x is given by, d[ln(2x)] / dx = 1/x, where the differentiation of ln2x is given with respect to the variable x. We can derive this formula using the chain rule method. The image given below shows the formula for the derivative of ln2x:

Now that we know that the derivative of ln2x is equal to 1/x, we will prove it using the chain rule method of differentiation. This method is used to find the derivative of the composite of functions. The formula for chain rule is d[f(g(x))] / dx = d[f(x)]/d[g(x)] × d[g(x)]/dx. Here, assume f(x) = ln x and g(x) = 2x, then we have f(g(x)) = ln2x. Also, we will use the following formulas to derive the derivative of ln2x:

• Derivative of lnx: d(ln x)/dx = 1/x
• Derivative of 2x: d(2x)/dx = 2

Using the above formulas, we have

d[ln(2x)] / dx = d[ln(2x)] / d(2x) × d(2x) / x --- [Using derivatives of composite functions formula]

= 1/(2x) × 2 --- [Because derivative of lnx is equal to 1/x and derivative of ax is equal to a.]

= 2/2x

= 1/x

Hence, we have proved that the derivative of ln2x is equal to 1/x using the chain rule formula.

We use logarithmic properties to simplify mathematical problems and determine the solution. We have various logarithmic properties. We will use the property of the ln of the product of ab, that is, ln(ab) = ln a + ln b. Here, consider a = 2 and b = x, then we have ln(2x) = ln2 + lnx. Now, we know that ln2 is a constant term and hence, its derivative is equal to zero and the derivative of ln x is equal to 1/x. Therefore, we have

d[ln(2x)] / dx = d[ln2 + lnx] / dx

= d(ln2) / dx + d(ln x)/dx

= 0 + 1/x

= 1/x

Hence, we have derived the formula for the derivative of ln2x using logarithmic properties.

In this section, we will evaluate the derivative of ln square x, that is, ln²x. To find the derivative of ln^2x, we will use the chain rule method, power rule of derivatives, and the derivative of lnx formula. So, using these formulas, we have

d[ln²x] / dx = 2 ln^2-1x × d(lnx)/dx

= 2 ln x × (1/x)

= (2 ln x) / x

Hence, the derivative of ln^2x is equal to (2 ln x) / x.

Important Notes on Derivative of ln2x

• The derivative of ln2x is equal to 1/x.
• We can determine the derivative of ln2x using the chain rule formula and logarithmic properties.
• The derivative of ln²x is equal to (2 ln x) / x.

☛ Related Topics:

• Log e
• Integration of logx
• Derivative of 2lnx