find the derivative of the function. f(x)= - 12x^(5/2)+6x^1.4 + x f(x)=□

Explanation:

Step1: Apply power - rule to first term

The power - rule for differentiation is $\frac{d}{dx}(ax^n)=nax^{n - 1}$. For the term $-12x^{\frac{5}{2}}$, we have $n=\frac{5}{2}$ and $a=-12$. So, $\frac{d}{dx}(-12x^{\frac{5}{2}})=\frac{5}{2}\times(-12)x^{\frac{5}{2}-1}=-30x^{\frac{3}{2}}$.

Step2: Apply power - rule to second term

For the term $6x^{1.4}$, with $n = 1.4$ and $a = 6$. Then $\frac{d}{dx}(6x^{1.4})=1.4\times6x^{1.4 - 1}=8.4x^{0.4}$.

Step3: Apply power - rule to third term

For the term $x=x^1$, with $n = 1$ and $a = 1$. So, $\frac{d}{dx}(x)=1\times1x^{1 - 1}=1$.

Step4: Sum up the derivatives

$f'(x)=\frac{d}{dx}(-12x^{\frac{5}{2}})+\frac{d}{dx}(6x^{1.4})+\frac{d}{dx}(x)=-30x^{\frac{3}{2}}+8.4x^{0.4}+1$.

Answer:

$-30x^{\frac{3}{2}}+8.4x^{0.4}+1$