d/dx[loga(x)]=
1/x(ln(a))
To find the derivative of loga(x), we will use the formula:
d/dx[loga(x)] = 1/(x * ln(a))
Where ln(a) is the natural logarithm of a.
Proof:
Let y = loga(x)
Then, ay = x (since a raised to the power of y gives x)
Now, we take the derivative of both sides with respect to x:
d(ay)/dx = d(x)/dx
Using the power rule of differentiation, we get:
[a * y’ * y] = 1
Simplifying, we get:
y’ = 1/(a * y)
Substituting y with loga(x), we get:
d/dx[loga(x)] = 1/(x * ln(a))
Therefore, the derivative of loga(x) is 1/(x * ln(a)).
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