# DERIVATIVE OF a TO THE POWER x

To find derivative of a function in which you have variable in exponent, you have to use logarithmic derivative. The following steps would be useful to do logarithmic derivative.

Lett y = f(x) be a function in which let the variable be in exponent.

Step 1 :

Take logarithm on both sides.

Step 2 :

Apply the power rule of logarithm.

Step 3 :

Find the derivative and solve for ᵈʸ⁄d.

Derivative of ax with respect to x.

Let y = ax.

In y = axwe have variable x in exponent.

y = ax

Take logarithm on both sides.

ln(y) = ln(ax)

Apply the power rule of logarithm on the right side.

ln(y) = xln(a)

Find the derivative with respect to x.

(Since a is a constant, ln(a) is also a constant. When we find derivative xln(a), we keep the the constant ln(a) as it is and find the derivative of x with respect to x, that is 1)

Multiply both sides by y.

Substitute y = ax.

Therefore, the derivative ax is axln(a).

Note :

Based on the derivative of ax, that is axlna, we can get the derivative of any constant to the power x.

Examples :

Derivative 2x = 2xln(2)

Derivative 3x = 3xln(3)

Derivative 5x = 5xln(5)

## Solved Problems

Find ᵈʸ⁄d.in each of the following.

Example 1 :

y = ex

(x and y are variables and e is a constant)

Solution :

In y = ex, we have constant e in base and variable x in exponent.

y = ex

Take logarithm on both sides.

ln(y) = ln(ex)

Apply the power rule of logarithm on the right side.

ln(y) = xln(e)

(The base of a natural logarithm is e, lne is a natural logarithm and its base is e)

ln(y) = xlnee

In a logarithm, if the base and argument are same, its value is 1. In lnee, the base and argument are same, so its value is 1.

ln(y) = x(1)

ln(y) = x

Find the derivative with respect to x.

Multiply both sides by y.

Substitute y = ex.

Example 2 :

y = 72x

Solution :

In y = 72x, we have variable in exponent.

y = 72x

Take logarithm on both sides.

ln(y) = ln(72x)

Apply the power rule of logarithm on the right side.

ln(y) = 2xln(7)

Find the derivative with respect to x.

Multiply both sides by y.

Substitute y = 72x.

Example 3 :

y = 3x

Solution :

In y = 3x, we have variable in exponent.

y = 3x

Take logarithm on both sides.

ln(y) = ln(3x)

Apply the power rule of logarithm on the right side.

ln(y) = xln(3)

Find the derivative with respect to x.

Multiply both sides by y.

Substitute y = 3x.

Example 4 :

y = 2ln(x)

Solution :

In y = 2ln(x), we have variable in exponent.

y = 2ln(x)

Take logarithm on both sides.

ln(y) = ln(2ln(x))

Apply the power rule of logarithm on the right side.

ln(y) = ln(x) ⋅ ln(2)

Find the derivative with respect to x.

Multiply both sides by y.

Substitute y = 2ln(x).

Example 5 :

y = 5sinx

Solution :

In y = 5sinx, we have variable in exponent.

y = 5sinx

Take logarithm on both sides.

ln(y) = ln(5sinx)

Apply the power rule of logarithm on the right side.

ln(y) = sinx ⋅ ln(5)

Find the derivative with respect to x.

Multiply both sides by y.

Substitute y = 5sinx.

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