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⁄dₓ.
Derivative of a^{x} with respect to x.
Let y = a^{x}.
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^{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)
Find ᵈʸ⁄dₓ.in each of the following.
Example 1 :
y = e^{x}
(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}.
Example 2 :
y = 7^{2x}
Solution :
In y = 7^{2x}, we have variable in exponent.
y = 7^{2x}
Take logarithm on both sides.
ln(y) = ln(7^{2x})
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 = 7^{2x}.
Example 3 :
y = 3^{√x}
Solution :
In y = 3^{√x}, we have variable in exponent.
y = 3^{√x}
Take logarithm on both sides.
ln(y) = ln(3^{√x})
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 = 3^{√x}.
Example 4 :
y = 2^{ln(x)}
Solution :
In y = 2^{ln(x)}, we have variable in exponent.
y = 2^{ln(x)}
Take logarithm on both sides.
ln(y) = ln(2^{ln(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 = 2^{ln(x)}.
Example 5 :
y = 5^{sinx}
Solution :
In y = 5^{sinx}, we have variable in exponent.
y = 5^{sinx}
Take logarithm on both sides.
ln(y) = ln(5^{sinx})
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 = 5^{sinx}.
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