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⁄dₓ.
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^{x}ln(a).
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}.
Note :
Based on the derivative of a^{x},and the above example 1, we can get the derivative of any constant to the power x.
Examples :
Derivative 3^{x} = 3^{x}ln(3)
Derivative 7^{x} = 7^{x}ln(7)
Derivative 13^{x} = 13^{x}ln(13)
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) = (x^{2} + 3x + 2)ln(6)
Find the derivative with respect to x.
Multiply both sides by y.
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