Derivative of 2lnx

The derivative of 2lnx is equal to 2/x. We can evaluate this derivative using the constant multiple rule of differentiation and the first principle of derivatives. Before getting into the detail about the derivative of 2lnx, let us recall the meaning of the derivative of a function. The derivative of a function gives the slope function of the tangent to the graph of the function. The derivative of 2lnx, which is 2/x, gives the slope of the tangent to the function f(x) = 2lnx at point x.

Further, in this article, we will calculate the derivative of 2lnx and derive its formula using different methods of differentiation. We will also evaluate the second derivative of 2lnx and derivatives of functions using its formula for a better understanding of the concept.

The derivative of 2lnx gives the rate of change of the function f(x) = 2lnx with respect to the change in the variable x as the differentiation of a function is a process of finding the rate of change of a function with respect to the change in the variable. We can write the derivative of 2lnx as d(2lnx)/dx (OR) (2lnx)' = 2/x. This derivative can be calculated using the formula for the derivative of ln x and the constant multiple rule of differentiation. We can also evaluate the derivative of 2lnx using the first principle of derivatives, that is, the definition of limits formula.

The formula for the derivative of 2lnx gives is given by, d(2lnx)/dx (OR) (2lnx)' = 2/x. 2/x is the rate of change function for f(x) = 2lnx with respect to the variable x. We can use different methods of differentiation to compute the derivative of 2lnx including:

• Constant Multiple Rule along with Derivative of Ln x Formula
• First Principle of Derivatives

As we know that the derivative of 2lnx is equal to 2/x, we will now prove this using the constant multiple rule whose formula is given by, d(k f(x))/dx = k d(f(x))/dx, where k is a constant. So, using the formula, we have k = 2 and f(x) = ln x. Also, we know that the derivative of ln x is equal to 1/x. Therefore, using the given formulas and facts, the derivative of 2 lnx is given by,

d(2 lnx)/dx = 2 × d(ln x)/dx

= 2 × (1/x) --- [Because d(ln x)/dx = 1/x]

= 2/x

Hence, we have derived the formula for the derivative of 2 lnx.

To prove that the derivative of 2lnx is equal to 2/x using the first principle of derivatives, we will use the following formulas:

• f'(x) = lim_h→0 [f(x + h) - f(x)] / h
• ln a - ln b = ln (a/b) → Logarithmic Product Rule
• L'hopital's Rule

Using the above formulas, we have

d(2lnx)/dx = lim_h→0 [2 ln (x + h) - 2 ln (x)] / h

= lim_h→0 2 [ln (x + h) - ln (x)] / h

= 2 lim_h→0 [ln (x + h) - ln (x)] / h

= 2 lim_h→0 [ln {(x + h)/x}] / h

= 2 lim_h→0 [ln (1 + h/x)] / h

Now, taking the limit gives the form 0/0 and if the limit gives the result of the form 0/0 or ∞/∞, then we use L'hopital's rule. To apply this rule, we take the derivative of the numerator and denominator with respect to the variable h separately and divide them. The derivative of ln (1 + h/x) is d(ln (1 + h/x))/dh = (1/x) / (1 + h/x) and the derivative of h is dh/dh = 1. Therefore, we have

2 lim_h→0 [ln (1 + h/x)] / h

= 2 lim_h→0 [d(ln (1 + h/x))/dh] / [dh/dh]

= 2 × lim_h→0 [(1/x) / (1 + h/x)] / 1

= 2 × lim_h→0 [1 / (x + h)]

= 2 × (1/x)

= 2/x

Hence, we have proved that the formula for the derivative of 2 lnx is equal to 2/x using the first principle of derivatives.

Important Notes on Derivative of 2lnx

• The derivative of 2lnx is equal to 2/x.
• We can evaluate the derivative of 2 lnx using different methods of derivatives including constant multiple rule and first principle.
• d(2lnx)/dx = 2/x

☛ Related Topics:

• Integration of log x
• Antilog table
• Log Table

What is Derivative of 2lnx in Math?

The derivative of 2lnx is equal to 2/x. It gives the slope of the tangent to the function f(x) = 2lnx at point x. The derivative of 2lnx implies the rate of change of the function f(x) = 2lnx with respect to the change in the variable x

What is the Formula for the Derivative of 2 lnx?

The formula for the derivative of 2lnx gives is given by, d(2lnx)/dx (OR) (2lnx)' = 2/x. We can determine this formula using the formula for the derivative of the logarithmic function.

How Do You Find the Derivative of 2lnx?

We can calculate the derivative of 2lnx using the constant multiple rule and derivative of lnx formula. We know that the derivative of ln x is equal to 1/x. Therefore, we have d(2lnx)/dx = 2 d(lnx)/dx = 2/x.

What is the Derivative of Sin²(lnx)?

The derivative of sin²(lnx) is given by, d(sin²(lnx))/dx = 2 sin(lnx) × d(sin(lnx))/dx × d(lnx)/dx = 2sin(lnx) × cos (lnx) × (1/x) = sin(2lnx)/x [Because sin2A = 2sinAcosA, derivative of sin x is cos x]. Therefore, derivative of sin²(lnx) is equal to sin(2lnx)/x.

What is the Second Derivative of 2lnx?

The second derivative of 2lnx is determined by differentiating the first derivative of 2lnx. Therefore, we have d²(2lnx)/dx² = d(2/x)/dx = -2/x². Hence, the second derivative of 2lnx is -2/x² .

Is the Derivative of 2lnx the Same as the Derivative of ln²x?

No, the derivative of 2lnx is not the same as the derivative of ln²x. Derivative of 2lnx is equal to 2/x whereas the derivative of ln²x is given by, d(ln²x)/dx = 2lnx × (1/x) = (2/x) lnx.

What is the Derivative of e^2lnx?

The derivative of e^2lnx is given by, d(e^2lnx)/dx = (2/x)e^2lnx. Therefore, the derivative of e^2lnx is equal to (2/x)e^2lnx .