Question

1. differentiate: $f(x)=(16x^{3}-5x^{6})^{5/2}$ $f(x)=$

Explanation:

Step1: Apply the chain - rule

Let $u = 16x^{3}-5x^{6}$, then $y = u^{\frac{5}{2}}$. The chain - rule states that $\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}$. First, find $\frac{dy}{du}$.
Using the power rule $\frac{d}{du}(u^{n})=nu^{n - 1}$, we have $\frac{dy}{du}=\frac{5}{2}u^{\frac{5}{2}-1}=\frac{5}{2}u^{\frac{3}{2}}$.

Step2: Find $\frac{du}{dx}$

Differentiate $u = 16x^{3}-5x^{6}$ with respect to $x$. Using the power rule $\frac{d}{dx}(ax^{n})=nax^{n - 1}$, we get $\frac{du}{dx}=16\times3x^{2}-5\times6x^{5}=48x^{2}-30x^{5}$.

Step3: Calculate $\frac{dy}{dx}$

Substitute $u = 16x^{3}-5x^{6}$ back into $\frac{dy}{du}$ and then multiply by $\frac{du}{dx}$:
$\frac{dy}{dx}=\frac{5}{2}(16x^{3}-5x^{6})^{\frac{3}{2}}(48x^{2}-30x^{5})$.
We can factor out a $6x^{2}$ from the second factor: $\frac{dy}{dx}=\frac{5}{2}(16x^{3}-5x^{6})^{\frac{3}{2}}\times6x^{2}(8 - 5x^{3})$.
Simplify to get $\frac{dy}{dx}=15x^{2}(8 - 5x^{3})(16x^{3}-5x^{6})^{\frac{3}{2}}$.

Answer:

$15x^{2}(8 - 5x^{3})(16x^{3}-5x^{6})^{\frac{3}{2}}$