Question

find the derivative of the following function.
y = 5x^{-\frac{3}{2}}+4x^{-\frac{1}{2}}+x^{11}-8
y=-\frac{15}{2}x^{-\frac{5}{2}}-2x^{-\frac{3}{2}}+11x^{10}

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 $5x^{-\frac{3}{2}}$, we have $n =-\frac{3}{2}$ and $a = 5$. So, $\frac{d}{dx}(5x^{-\frac{3}{2}})=5\times(-\frac{3}{2})x^{-\frac{3}{2}-1}=-\frac{15}{2}x^{-\frac{5}{2}}$.

Step2: Apply power - rule to second term

For the term $4x^{-\frac{1}{2}}$, with $n =-\frac{1}{2}$ and $a = 4$. Then $\frac{d}{dx}(4x^{-\frac{1}{2}})=4\times(-\frac{1}{2})x^{-\frac{1}{2}-1}=-2x^{-\frac{3}{2}}$.

Step3: Apply power - rule to third term

For the term $x^{11}$, with $n = 11$ and $a = 1$. So, $\frac{d}{dx}(x^{11})=11x^{10}$.

Step4: Differentiate the constant term

The derivative of a constant $C$ is 0. Since $C=-8$, $\frac{d}{dx}(-8)=0$.

Step5: Combine the derivatives

$y'=-\frac{15}{2}x^{-\frac{5}{2}}-2x^{-\frac{3}{2}}+11x^{10}+0=-\frac{15}{2}x^{-\frac{5}{2}}-2x^{-\frac{3}{2}}+11x^{10}$

Answer:

$y'=-\frac{15}{2}x^{-\frac{5}{2}}-2x^{-\frac{3}{2}}+11x^{10}$