Question

Differentiate the function.

\(k\left( r \right) = {e^r} + {r^e}\)

Short answer

Derivative of the above function is \({e^r} + e{r^{e - 1}}\).

Step-by-step solution

Precise Definition of differentiation

A Derivative is used in calculus mathematics.It is the rate of change of a function with respect to a variable. A derivative is a function of a single variable at a selected input value.

Sum rule of differentiation

Use the rule of differentiation which can apply to this question:

\(\frac{d}{{dx}}\left( {f\left( x \right) + g\left( x \right)} \right) = \frac{d}{{dx}}f\left( x \right) + \frac{d}{{dx}}g\left( x \right)\).

\(\begin{aligned}\frac{d}{{dr}}\left( {k\left( r \right)} \right) &= \frac{d}{{dr}}\left( {{e^r} + {r^e}} \right)\\{k'}\left( r \right) &= \frac{d}{{dr}}\left( {{e^r}} \right) + \frac{d}{{dr}}\left( {{r^e}} \right)\end{aligned}\)

Power rule of differentiation

The power rule of differentiation is that \(\frac{d}{{dx}}\left( {{x^n}} \right) = n{x^{n - 1}}\), where \(n\) is any real number.

Apply power rule and simplify.

\(k'\left( r \right) = {e^r} + e{r^{e - 1}}\)

Therefore, the differentiation of the function is \(k'\left( r \right) = {e^r} + e{r^{e - 1}}\).