Given the function f(x)=(-9-5x^(3))(7x+2x^(-3)+2), find f^(')(x) in any form. Answer: f^(')(x)=

Apply Product Rule: First, we need to apply the product rule to find the derivative of the function f(x). The product rule states that the derivative of a product of two functions is the derivative of the first function times the second function plus the first function times the derivative of the second function.Define Functions: Let's denote the two functions as u(x) = -9 - 5x^3 and v(x) = 7x + 2x^(-3) + 2. We need to find the derivatives u'(x) and v'(x).Find Derivatives: The derivative of u(x) = -9 - 5x^3 with respect to x is u'(x) = -15x^2.Apply Product Rule: The derivative of v(x) = 7x + 2x^(-3) + 2 with respect to x is v'(x) = 7 - 6x^(-4).Substitute Expressions: Now we apply the product rule: f'(x) = u'(x)v(x) + u(x)v'(x).Simplify Expression: Substitute the expressions for u'(x), v(x), u(x), and v'(x) into the product rule formula: f'(x) = (-15x^2)(7x + 2x^(-3) + 2) + (-9 - 5x^3)(7 - 6x^(-4)).Combine Like Terms: Simplify the expression by multiplying the terms: f'(x) = -105x^3 - 30x^(-1) - 30x^2 - 63x - 10x^(-3) + 54x^(-4).Combine like terms to get the final derivative in any form: f'(x) = -105x^3 - 30x^2 - 63x - 30x^(-1) - 10x^(-3) + 54x^(-4).

Resources

Testimonials

Plans

Resources

Testimonials

Plans

AI tutor

Welcome to Bytelearn!

Let’s check out your problem:

Given the function
f(x)=(-9-5x^(3))(7x+2x^(-3)+2), find
f^(')(x) in any form.
Answer:
f^(')(x)=

Given the function $f(x)=\left(-9-5 x^{3}\right)\left(7 x+2 x^{-3}+2\right)$ , find $f^{\prime}(x)$ in any form. $\newline$ Answer: $f^{\prime}(x)=$

View step-by-step help

Home

Math Problems

Algebra 2

Composition of linear and quadratic functions: find an equation

Full solution

Q. Given the function

f(x)=\left(-9-5 x^{3}\right)\left(7 x+2 x^{-3}+2\right)

, find

f^{\prime}(x)

in any form.

\newline

Answer:

f^{\prime}(x)=

Apply Product Rule: First, we need to apply the product rule to find the derivative of the function $f(x)$ . The product rule states that the derivative of a product of two functions is the derivative of the first function times the second function plus the first function times the derivative of the second function.
Define Functions: Let's denote the two functions as $u(x) = -9 - 5x^3$ and $v(x) = 7x + 2x^{-3} + 2$ . We need to find the derivatives $u'(x)$ and $v'(x)$ .
Find Derivatives: The derivative of $u(x) = -9 - 5x^3$ with respect to $x$ is $u'(x) = -15x^2$ .
Apply Product Rule: The derivative of $v(x) = 7x + 2x^{-3} + 2$ with respect to $x$ is $v'(x) = 7 - 6x^{-4}$ .
Substitute Expressions: Now we apply the product rule: $f'(x) = u'(x)v(x) + u(x)v'(x)$ .
Simplify Expression: Substitute the expressions for $u'(x)$ , $v(x)$ , $u(x)$ , and $v'(x)$ into the product rule formula: $f'(x) = (-15x^2)(7x + 2x^{-3} + 2) + (-9 - 5x^3)(7 - 6x^{-4})$ .
Combine Like Terms: Simplify the expression by multiplying the terms: $f'(x) = -105x^3 - 30x^{-1} - 30x^2 - 63x - 10x^{-3} + 54x^{-4}$ .
Combine Like Terms: Simplify the expression by multiplying the terms: $f'(x) = -105x^3 - 30x^{-1} - 30x^2 - 63x - 10x^{-3} + 54x^{-4}$ .Combine like terms to get the final derivative in any form: $f'(x) = -105x^3 - 30x^2 - 63x - 30x^{-1} - 10x^{-3} + 54x^{-4}$ .