Given the function y=(3x^(-3)+9)(-6-2x-7x^(3)), find (dy)/(dx) in any form. Answer: (dy)/(dx)=

Apply Product Rule: Apply the product rule to find the derivative of the product of two functions. The product rule states that (d/dx)[u(x)v(x)] = u'(x)v(x) + u(x)v'(x), where u(x) and v(x) are functions of x. Let u(x) = 3x^(-3) + 9 and v(x) = -6 - 2x - 7x^(3).Find u'(x): Find the derivative of u(x) with respect to x. u'(x) = d/dx[3x^(-3) + 9] = d/dx[3x^(-3)] + d/dx[9] u'(x) = -9x^(-4) + 0 u'(x) = -9x^(-4)Find v'(x): Find the derivative of v(x) with respect to x. v'(x) = d/dx[-6 - 2x - 7x^(3)] = d/dx[-6] - d/dx[2x] - d/dx[7x^(3)] v'(x) = 0 - 2 - 21x^(2) v'(x) = -2 - 21x^(2)Apply Product Rule with Derivatives: Apply the product rule using the derivatives u'(x) and v'(x). (dy)/(dx) = u'(x)v(x) + u(x)v'(x) (dy)/(dx) = (-9x^(-4))(-6 - 2x - 7x^(3)) + (3x^(-3) + 9)(-2 - 21x^(2))Simplify Expression: Simplify the expression by distributing and combining like terms. (dy)/(dx) = 54x^(-4) + 18x^(-3) + 63x^(-1) - 6x^(-3) - 18 - 63x^(2) (dy)/(dx) = 54x^(-4) + (18x^(-3) - 6x^(-3)) + 63x^(-1) - 18 - 63x^(2) (dy)/(dx) = 54x^(-4) + 12x^(-3) + 63x^(-1) - 18 - 63x^(2)

Resources

Testimonials

Plans

Resources

Testimonials

Plans

AI tutor

Welcome to Bytelearn!

Let’s check out your problem:

Given the function
y=(3x^(-3)+9)(-6-2x-7x^(3)), find
(dy)/(dx) in any form.
Answer:
(dy)/(dx)=

Given the function $y=\left(3 x^{-3}+9\right)\left(-6-2 x-7 x^{3}\right)$ , find $\frac{d y}{d x}$ in any form. $\newline$ Answer: $\frac{d y}{d x}=$

View step-by-step help

Home

Math Problems

Algebra 1

Transformations of quadratic functions

Full solution

Q. Given the function

y=\left(3 x^{-3}+9\right)\left(-6-2 x-7 x^{3}\right)

, find

\frac{d y}{d x}

in any form.

\newline

Answer:

\frac{d y}{d x}=

Apply Product Rule: Apply the product rule to find the derivative of the product of two functions. $\newline$ The product rule states that $(\frac{d}{dx})[u(x)v(x)] = u'(x)v(x) + u(x)v'(x)$ , where $u(x)$ and $v(x)$ are functions of $x$ . $\newline$ Let $u(x) = 3x^{-3} + 9$ and $v(x) = -6 - 2x - 7x^{3}$ .
Find $u'(x)$ : Find the derivative of $u(x)$ with respect to $x$ .
$u'(x) = \frac{d}{dx}[3x^{-3} + 9] = \frac{d}{dx}[3x^{-3}] + \frac{d}{dx}[9]$
$u'(x) = -9x^{-4} + 0$
$u'(x) = -9x^{-4}$
Find $v'(x)$ : Find the derivative of $v(x)$ with respect to $x$ .
$v'(x) = \frac{d}{dx}[-6 - 2x - 7x^{3}] = \frac{d}{dx}[-6] - \frac{d}{dx}[2x] - \frac{d}{dx}[7x^{3}]$
$v'(x) = 0 - 2 - 21x^{2}$
$v'(x) = -2 - 21x^{2}$
Apply Product Rule with Derivatives: Apply the product rule using the derivatives $u'(x)$ and $v'(x)$ . $\newline$ $\frac{dy}{dx} = u'(x)v(x) + u(x)v'(x)$ $\newline$ $\frac{dy}{dx} = (-9x^{-4})(-6 - 2x - 7x^{3}) + (3x^{-3} + 9)(-2 - 21x^{2})$
Simplify Expression: Simplify the expression by distributing and combining like terms. $\newline$ $(dy)/(dx) = 54x^{-4} + 18x^{-3} + 63x^{-1} - 6x^{-3} - 18 - 63x^{2}$ $\newline$ $(dy)/(dx) = 54x^{-4} + (18x^{-3} - 6x^{-3}) + 63x^{-1} - 18 - 63x^{2}$ $\newline$ $(dy)/(dx) = 54x^{-4} + 12x^{-3} + 63x^{-1} - 18 - 63x^{2}$