1. find the derivative of each function\na) d/dx (x³ tan⁻¹(5x))

Explanation:

Step1: Apply product - rule

The product - rule states that if $y = u\cdot v$, then $y^\prime=u^\prime v + uv^\prime$. Let $u = x^{3}$ and $v=\tan^{- 1}(5x)$.

Step2: Find $u^\prime$

Differentiate $u = x^{3}$ with respect to $x$. Using the power - rule $\frac{d}{dx}(x^{n})=nx^{n - 1}$, we have $u^\prime=\frac{d}{dx}(x^{3}) = 3x^{2}$.

Step3: Find $v^\prime$

Differentiate $v=\tan^{-1}(5x)$ with respect to $x$. Using the formula $\frac{d}{dx}(\tan^{-1}(ax))=\frac{a}{1+(ax)^{2}}$, here $a = 5$, so $v^\prime=\frac{5}{1 + 25x^{2}}$.

Step4: Apply the product - rule formula

$y^\prime=u^\prime v+uv^\prime=3x^{2}\tan^{-1}(5x)+x^{3}\cdot\frac{5}{1 + 25x^{2}}$.

Answer:

$3x^{2}\tan^{-1}(5x)+\frac{5x^{3}}{1 + 25x^{2}}$