Partial Derivatives Example 4

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.

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:

\[\frac{{\partial f}}{{\partial y}} = \frac{{\frac{{2y}}{{2\sqrt {{x^2} + {y^2}} }}}}{{x + \sqrt {{x^2} + {y^2}} }}\]

Please see our other examples on partial derivatives, including chain rule, product rule, quotient rule, and multivariable implicit differentiation:

Partial Derivatives Examples:

Partial Derivatives (Ex1)

Partial Derivatives (Ex2)

Partial Derivatives (Ex3)

Multivariable Chain Rule Examples:

Multivariable Chain Rule (Ex1)

Multivariable Chain Rule (Ex2)

Implicit Multivariable Differentiation:

Multivariable Implicit Differentiation

Return to the multivariable differentiation hub page:

Partial Derivatives, Multivariable Differentiation

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