Solve k = omega/c

Simplify

k = \frac{ω}{c}

Find the first partial derivative by treating the variable c as a constant and differentiating with respect to ω

\frac{\partial k}{\partial ω} = \frac{\partial}{\partial ω} (\frac{ω}{c})

Use differentiation rule \frac{\partial}{\partial x} (\frac{f ( x )}{g ( x )}) = \frac{\frac{\partial}{\partial x} ( f ( x )) \times g ( x ) - f ( x ) \times \frac{\partial}{\partial x} ( g ( x ) )}{( g ( x ) ) ^{2}}

\frac{\partial k}{\partial ω} = \frac{\frac{\partial}{\partial ω} ( ω ) c - ω \times \frac{\partial}{\partial ω} ( c )}{c ^{2}}

Use \frac{\partial}{\partial x} x^{n} = n x^{n - 1} to find derivative

\frac{\partial k}{\partial ω} = \frac{1 \times c - ω \times \frac{\partial}{\partial ω} ( c )}{c ^{2}}

Use \frac{\partial}{\partial x} (c) = 0 to find derivative

\frac{\partial k}{\partial ω} = \frac{1 \times c - ω \times 0}{c ^{2}}

Any expression multiplied by 1 remains the same

\frac{\partial k}{\partial ω} = \frac{c - ω \times 0}{c ^{2}}

Any expression multiplied by 0 equals 0

\frac{\partial k}{\partial ω} = \frac{c - 0}{c ^{2}}

Removing 0 doesn't change the value,so remove it from the expression

\frac{\partial k}{\partial ω} = \frac{c}{c ^{2}}

Solution

More Steps

\frac{\partial k}{\partial ω} = \frac{1}{c}

Function