Question

differentiate: $f(x)=(17x^{3}-5x^{0})^{5/2}$
$f(x)=$

Explanation:

Step1: Identify the outer - inner functions

Let $u = 17x^{3}-5x^{0}=17x^{3}-5$ (since $x^{0}=1$ for $x
eq0$), and $y = u^{\frac{5}{2}}$.

Step2: Differentiate the outer function

Using the power rule $\frac{d}{du}(u^{n})=nu^{n - 1}$, we have $\frac{dy}{du}=\frac{5}{2}u^{\frac{5}{2}-1}=\frac{5}{2}u^{\frac{3}{2}}$.

Step3: Differentiate the inner function

$\frac{du}{dx}=\frac{d}{dx}(17x^{3}-5)=17\times3x^{2}-0 = 51x^{2}$ (using the power rule $\frac{d}{dx}(ax^{n})=nax^{n - 1}$ and $\frac{d}{dx}(c)=0$ for a constant $c$).

Step4: Apply the chain - rule

The chain - rule states that $\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}$. Substituting $u = 17x^{3}-5$, $\frac{dy}{du}=\frac{5}{2}u^{\frac{3}{2}}$, and $\frac{du}{dx}=51x^{2}$, we get $\frac{dy}{dx}=\frac{5}{2}(17x^{3}-5)^{\frac{3}{2}}\cdot51x^{2}=\frac{255}{2}x^{2}(17x^{3}-5)^{\frac{3}{2}}$.

Answer:

$\frac{255}{2}x^{2}(17x^{3}-5)^{\frac{3}{2}}$