Question

\(d\left(e^{5 x}+x^{5}\right)\)

Short answer

The derivative \(d\left(e^{5 x}+x^{5}\right)\) is \(5e^{5x} + 5x^{4}\).

Step-by-step solution

Differentiate \(e^{5x}\)

The derivative of \(e^{u}\) is \(e^{u} \cdot u'\), where \(u'\) is the derivative of \(u\). In this case, \(u = 5x\), so \(u' = 5\). Therefore, the derivative of \(e^{5x}\) is \(5e^{5x}\).

Differentiate \(x^{5}\)

The power rule states that the derivative of \(x^n\) is \(nx^{n-1}\). Here, \(n = 5\), so the derivative of \(x^5\) is \(5x^{4}\).

Combine the two derivatives

The derivative of the sum of two functions is the sum of their derivatives. In this case, the derivative of \(e^{5x}\) is \(5e^{5x}\) and the derivative of \(x^{5}\) is \(5x^{4}\). Hence, the derivative of \(e^{5 x}+x^{5}\) is \(5e^{5x} + 5x^{4}\).