# Use the product rule to find the derivative. Y = (5x^2 + 3)(2x - 5).

## Question:

Use the product rule to find the derivative.

{eq}Y = (5x^2 + 3)(2x - 5) {/eq}

## Differentiation:

The differentiation is an important concept used in various theories like statistics and mathematics as it computes the rate of change in the value of the function. It is used in computing the differential equations and the distribution function of the random variable.

## Answer and Explanation:

The derivative of the following expression can be found using product rule as:

{eq}\begin{align*} Y &= \left( {5{x^2} + 3} \right)\left( {2x - 5} \right)\\ \dfrac{{dY}}{{dx}} &= \left( {5{x^2} + 3} \right)\left[ {\dfrac{d}{{dx}}\left\{ {\left( {2x - 5} \right)} \right\}} \right] + \left( {2x - 5} \right)\left[ {\dfrac{d}{{dx}}\left\{ {\left( {5{x^2} + 3} \right)} \right\}} \right]\\ &= \left( {5{x^2} + 3} \right)\left[ 2 \right] + \left( {2x - 5} \right)\left[ {10x} \right]\\ &= 10{x^2} + 6 + 20{x^2} - 50x\\ &= 30{x^2} - 50x + 6\\ &= 2\left( {15{x^2} - 25x + 3} \right) \end{align*} {/eq}

Using Differentiation to Find Maximum and Minimum Values

from Math 104: Calculus

Chapter 9 / Lesson 4
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