Derivative of a Constant to the Power of x

Derivative of a constant to the power of x.

Consider the following function :

y = a^x

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

In y = a^x, we have variable x in exponent.

y = a^x

Take logarithm on both sides.

ln(y) = ln(a^x)

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 = a^x.

Therefore, the derivative a^x is a^xln(a).

Solved Problems

Find ᵈʸ⁄dₓ.in each of the following.

Example 1 :

y = 2^x

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

Solution :

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

y = 2^x

Take logarithm on both sides.

ln(y) = ln(2^x)

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 = 2^x.

Example 2 :

y = 5^3x

Solution :

In y = 5^3x, we have variable in exponent.

y = 5^3x

Take natural logarithm on both sides.

ln(y) = ln(5^3x)

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 = 5^3x.

Example 3 :

y = 8^√x

Solution :

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

y = 8^√x

Take natural logarithm on both sides.

ln(y) = ln(8^√x)

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 = 3^√x.

Example 4 :

y = 10^ln(x)

Solution :

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

y = 10^ln(x)

Take natural logarithm on both sides.

ln(y) = ln(10^ln(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 = 10^ln(x).

Example 5 :

y = 2^tanx

Solution :

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

y = 2^tanx

Take natural logarithm on both sides.

ln(y) = ln(2^tanx)

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 = 2^tanx.

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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