Solve a_c = \frac{v^2}{r}

Simplify

a_{c} = \frac{v ^{2}}{r}

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

\frac{\partial a _{c}}{\partial v} = \frac{\partial}{\partial v} (\frac{v ^{2}}{r})

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 a _{c}}{\partial v} = \frac{\frac{\partial}{\partial v} ( v ^{2} ) r - v ^{2} \times \frac{\partial}{\partial v} ( r )}{r ^{2}}

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

\frac{\partial a _{c}}{\partial v} = \frac{2 v r - v ^{2} \times \frac{\partial}{\partial v} ( r )}{r ^{2}}

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

\frac{\partial a _{c}}{\partial v} = \frac{2 v r - v ^{2} \times 0}{r ^{2}}

Any expression multiplied by 0 equals 0

\frac{\partial a _{c}}{\partial v} = \frac{2 v r - 0}{r ^{2}}

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

\frac{\partial a _{c}}{\partial v} = \frac{2 v r}{r ^{2}}

Solution

More Steps

\frac{\partial a _{c}}{\partial v} = \frac{2 v}{r}

Function