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). Assume there is a variable in exponent in f(x).
Step 1 :
Take logarithm on both sides.
Step 2 :
Apply the power rule of logarithm.
Step 3 :
Find the derivative and solve for ᵈʸ⁄dₓ.
(if required, the product rule of derivative can be used)
Derivative of x^{x} with respect to x.
Let y = x^{x}.
In y = x^{x}, we have variable x in exponent.
y = x^{x}
Take logarithm on both sides.
ln(y) = ln(x^{x})
Use the Power Rule of Logarithm on the right side.
ln(y) = xln(x)
Now, we have to find the derivative on both sides with respect to x.
To find the derivative of ln(y) with respect to x, we have to use chain rule and to find the derivative of xlnx with respect to x, we have to use product rule.
Multiply both sides by y.
Substitute y = x^{x}.
Therefore, the derivative x^{x} is x^{x}(1 + ln(x)).
Problem 1 :
Find the derivative of a^{x} with respect to x.
or
Given y = a^{x}, find ᵈʸ⁄dₓ.
(x and y are variables and a is a constant)
Solution :
In y = a^{x}, we have constant a in base and 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, lna is also a constant. When we find derivative xlna, we keep the the constant lna 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^{x}ln(a).
Note :
Based on the derivative of a^{x}, that is a^{x}lna, we can get the derivative of any constant to the power x.
Examples :
Derivative 2^{x} = 2^{x}ln(2)
Derivative 3^{x} = 3^{x}ln(3)
Derivative 5^{x} = 5^{x}ln(5)
Problem 2 :
Find the derivative of e^{x} with respect to x.
or
Given y = e^{x}, find ᵈʸ⁄dₓ.
(x and y are variables and e is a constant)
Solution :
In y = e^{x}, we have constant e in base and variable x in exponent.
y = e^{x}
Take logarithm on both sides.
ln(y) = ln(e^{x})
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) = xln_{e}e
In a logarithm, if the base and argument are same, its value is 1. In ln_{e}e, 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 = e^{x}.
Therefore, the derivative e^{x} is e^{x}.
Problem 3 :
Find the derivative of a^{a} with respect to x.
or
Given y = a^{a}, find ᵈʸ⁄dₓ.
(y is a variable and a is a constant)
Solution :
In y = a^{a}, both base and exponent, we have the constant a.
Since a is a constant, a^{a} is also a constant.
y = a^{a}
ᵈʸ⁄dₓ = 0
Since the derivative of a constant is zero, the derivative of a^{a }is zero.
Problem 4 :
Find the derivative of a^{2x} with respect to x.
or
Given y = a^{2x}, find ᵈʸ⁄dₓ.
Solution :
In y = a^{2x}, we have a constant in base and variable in exponent.
y = a^{2x}
Take logarithm on both sides.
ln(y) = ln(a^{2x})
Apply the power rule of logarithm on the right side.
ln(y) = 2x ⋅ ln(a)
Find the derivative with respect to x.
Multiply both sides by y.
Substitute y = a^{2x}.
Therefore, the derivative a^{2x} is 2a^{2x}ln(a).
Problem 5 :
Find the derivative of a^{√x} with respect to x.
or
Given y = a^{√x}, find ᵈʸ⁄dₓ.
Solution :
In y = a^{2x}, we have a constant in base and variable 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) = √x ⋅ ln(a)
ln(y) = x^{½} ⋅ ln(a)
Find the derivative with respect to x.
Multiply both sides by y.
Substitute y = a^{√x}.
Therefore, the derivative y = a^{√x} is
Problem 6 :
Find the derivative of a^{ln(x)} with respect to x.
or
Given y = a^{ln(x)}, find ᵈʸ⁄dₓ.
(y is a variable and a is a constant)
Solution :
In y = a^{ln(x)}, we have a constant in base and variable in exponent.
y = a^{ln(x)}
Take logarithm on both sides.
ln(y) = ln(a^{ln(x)})
Apply the power rule of logarithm on the right side.
ln(y) = ln(x) ⋅ ln(a)
Find the derivative with respect to x.
Multiply both sides by y.
Substitute y = a^{ln(x)}.
Therefore, the derivative a^{ln(x)} is
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