timed problem score: 0/10 current time: 1.3 d/dx (1/5 x^4) answer 3/5 x^3 4/5 x^3 1/25 x^5 1/20 x^5 keyboard…

Explanation:

Step1: Apply constant - multiple rule

The constant - multiple rule of differentiation states that if (y = cf(x)), then (y'=c\cdot f'(x)), where (c=\frac{1}{5}) and (f(x)=x^{4}). So (\frac{d}{dx}(\frac{1}{5}x^{4})=\frac{1}{5}\cdot\frac{d}{dx}(x^{4})).

Step2: Apply power rule

The power rule of differentiation is (\frac{d}{dx}(x^{n}) = nx^{n - 1}). For (n = 4), (\frac{d}{dx}(x^{4})=4x^{3}).

Step3: Calculate the result

Substitute (\frac{d}{dx}(x^{4}) = 4x^{3}) into (\frac{1}{5}\cdot\frac{d}{dx}(x^{4})), we get (\frac{1}{5}\times4x^{3}=\frac{4}{5}x^{3}).

Answer:

(\frac{4}{5}x^{3})