Derivative of a Constant to the Power of x

Working rule to to find the derivative of a function which contains variable in exponent

Step 1 :

Take natural logarithm on both sides.

Step 2 :

Execute the power rule of logarithm.

Step 3 :

Find the derivative on both sides with respect to x and solve for ᵈʸ⁄d.

Derivative of a constant to the power of x.

Consider the following function :

y = ax

where x and y are variables and a is a constant.

In y = ax, we 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).

Solved Problems

Find ᵈʸ⁄d.in each of the following.

Example 1 :

y = 2x

(x and y are variables and 2 is a constant)

Solution :

In y = 2x, we have constant e in base and variable x in exponent.

y = 2x

Take logarithm on both sides.

ln(y) = ln(2x)

Apply the power rule of logarithm on the right side.

ln(y) = xln(2)

Find the derivative on both sides with respect to x.

Multiply both sides by y.

Substitute y = 2x.

Note :

Based on the derivative of ax,and the above example 1, we can get the derivative of any constant to the power x.

Examples :

Derivative 3x = 3xln(3)

Derivative 7x = 7xln(7)

Derivative 13x = 13xln(13)

Example 2 :

y = 53x

Solution :

In y = 53x, we have variable in exponent.

y = 53x

Take natural logarithm on both sides.

ln(y) = ln(53x)

Apply the power rule of logarithm on the right side.

ln(y) = 3xln(5)

Find the derivative on both sides with respect to x.

Multiply both sides by y.

Substitute y = 53x.

Example 3 :

y = 8x

Solution :

In y = 8x, we have variable in exponent.

y = 8x

Take natural logarithm on both sides.

ln(y) = ln(8x)

Apply the power rule of logarithm on the right side.

ln(y) = xln(8)

Find the derivative on both sides with respect to x.

Multiply both sides by y.

Substitute y = 3x.

Example 4 :

y = 10ln(x)

Solution :

In y = 10ln(x), we have variable in exponent.

y = 10ln(x)

Take natural logarithm on both sides.

ln(y) = ln(10ln(x))

Apply the power rule of logarithm on the right side.

ln(y) = ln(x) ⋅ ln(10)

Find the derivative with respect to x.

Multiply both sides by y.

Substitute y = 10ln(x).

Example 5 :

y = 2tanx

Solution :

In y = 2tanx, we have variable in exponent.

y = 2tanx

Take natural logarithm on both sides.

ln(y) = ln(2tanx)

Apply the power rule of logarithm on the right side.

ln(y) = tanx ⋅ ln(2)

Find the derivative with respect to x.

Multiply both sides by y.

Substitute y = 2tanx.

Example 6 :

Solution :

Take natural logarithm on both sides.

Apply the power rule of logarithm on the right side.

ln(y) = (x2 + 3x + 2)ln(6)

Find the derivative with respect to x.

Multiply both sides by y.

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