Home»Math Guides»Partial Derivatives (logarithmic function) – Example 4
Partial Derivatives Example 4 (partial derivatives of a natural logarithm)
In multivariable calculus you may be asked to find the partial derivatives.
When deriving with respect to a variable, just treat all other variables as a constant.
Let’s try an example where we find partial derivatives of a natural logarithm function, which is just shortened as ln.
Example problem: Find all first partial derivatives of \(f(x,y) = \ln \left( {x + \sqrt {{x^2} + {y^2}} } \right)\)
We just differentiate this like any other ln function, except we do it with respect to one of the variables at a time.
To differentiate a ln function, use this formula:
\[{\left[ {\ln (f)} \right]^\prime } = \frac{{f’}}{f}\]
Which just means differentiate what’s inside the ln and put that over what was inside originally.
Let’s differentiate with respect to x first.
Remember that the square root is the same as a 1/2 power, and differentiating it turns it into a -1/2 power, which can then just be put as a square root in a denominator.
\[\frac{{\partial f}}{{\partial x}} = \frac{{1 + \frac{{2x}}{{2\sqrt {{x^2} + {y^2}} }}}}{{x + \sqrt {{x^2} + {y^2}} }}\]
Which is basically the differentiation of what’s inside the ln with respect to x, over what was originally inside the ln.
Now let’s differentiate with respect to y:
\[\frac{{\partial f}}{{\partial y}} = \frac{{\frac{{2y}}{{2\sqrt {{x^2} + {y^2}} }}}}{{x + \sqrt {{x^2} + {y^2}} }}\]
Click here to move to another topic, partial derivatives where you use the multivariable chain rule!