Resourcesdown chevron
Testimonials
Plans
Sign in
Sign up
Resourcesright arrow
Testimonials
Plans
Bytelearn - cat image with glassesAI tutor

Welcome to Bytelearn!

Let’s check out your problem:

Given the function f(x)=(-5x^(3)-8x^(2)-1)(9x^(2)+1), find f^(')(x) in any form. Answer: f^(')(x)=

Given the function f(x)=(−5x3−8x2−1)(9x2+1) f(x)=\left(-5 x^{3}-8 x^{2}-1\right)\left(9 x^{2}+1\right) , find f′(x) f^{\prime}(x) in any form.\newlineAnswer: f′(x)= f^{\prime}(x)=

View step-by-step helpView step-by-step help
Homeright chevron icon
Math Problemsright chevron icon
Algebra 1right chevron icon
Transformations of quadratic functionsright chevron icon

Full solution

Q. Given the function f(x)=(−5x3−8x2−1)(9x2+1) f(x)=\left(-5 x^{3}-8 x^{2}-1\right)\left(9 x^{2}+1\right) , find f′(x) f^{\prime}(x) in any form.\newlineAnswer: f′(x)= f^{\prime}(x)=
  1. Product Rule Explanation: To find the derivative of the product of two functions, we use the product rule. 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.
  2. Define Functions: Let's denote the first function as u(x)=−5x3−8x2−1u(x) = -5x^3 - 8x^2 - 1 and the second function as v(x)=9x2+1v(x) = 9x^2 + 1. We need to find the derivatives u′(x)u'(x) and v′(x)v'(x).
  3. Find Derivatives: First, we find the derivative of u(x)u(x). The derivative of −5x3-5x^3 is −15x2-15x^2, the derivative of −8x2-8x^2 is −16x-16x, and the derivative of a constant −1-1 is 00. So, u′(x)=−15x2−16xu'(x) = -15x^2 - 16x.
  4. Apply Product Rule: Next, we find the derivative of v(x)v(x). The derivative of 9x29x^2 is 18x18x, and the derivative of a constant 11 is 00. So, v′(x)=18xv'(x) = 18x.
  5. Expand Expressions: Now we apply the product rule: f′(x)=u′(x)v(x)+u(x)v′(x)f'(x) = u'(x)v(x) + u(x)v'(x). Substituting the derivatives we found, we get f′(x)=(−15x2−16x)(9x2+1)+(−5x3−8x2−1)(18x)f'(x) = (-15x^2 - 16x)(9x^2 + 1) + (-5x^3 - 8x^2 - 1)(18x).
  6. Combine Like Terms: We need to expand the expressions to simplify them. First, we expand (−15x2−16x)(9x2+1)(-15x^2 - 16x)(9x^2 + 1), which gives us −135x4−15x2−144x3−16x-135x^4 - 15x^2 - 144x^3 - 16x.
  7. Final Derivative: Next, we expand (−5x3−8x2−1)(18x)(-5x^3 - 8x^2 - 1)(18x), which gives us −90x4−144x3−18x-90x^4 - 144x^3 - 18x.
  8. Final Derivative: Next, we expand (−5x3−8x2−1)(18x)(-5x^3 - 8x^2 - 1)(18x), which gives us −90x4−144x3−18x-90x^4 - 144x^3 - 18x.Now we combine like terms from both expansions. The terms −135x4-135x^4 and −90x4-90x^4 combine to −225x4-225x^4. The terms −144x3-144x^3 and −144x3-144x^3 combine to −288x3-288x^3. The terms −15x2-15x^2 and −8x2-8x^2 combine to −90x4−144x3−18x-90x^4 - 144x^3 - 18x00. The terms −90x4−144x3−18x-90x^4 - 144x^3 - 18x11 and −90x4−144x3−18x-90x^4 - 144x^3 - 18x22 combine to −90x4−144x3−18x-90x^4 - 144x^3 - 18x33.
  9. Final Derivative: Next, we expand (−5x3−8x2−1)(18x)(-5x^3 - 8x^2 - 1)(18x), which gives us −90x4−144x3−18x-90x^4 - 144x^3 - 18x.Now we combine like terms from both expansions. The terms −135x4-135x^4 and −90x4-90x^4 combine to −225x4-225x^4. The terms −144x3-144x^3 and −144x3-144x^3 combine to −288x3-288x^3. The terms −15x2-15x^2 and −8x2-8x^2 combine to −90x4−144x3−18x-90x^4 - 144x^3 - 18x00. The terms −90x4−144x3−18x-90x^4 - 144x^3 - 18x11 and −90x4−144x3−18x-90x^4 - 144x^3 - 18x22 combine to −90x4−144x3−18x-90x^4 - 144x^3 - 18x33.The final derivative −90x4−144x3−18x-90x^4 - 144x^3 - 18x44 is then −90x4−144x3−18x-90x^4 - 144x^3 - 18x55.

More problems from Transformations of quadratic functions

Question
Solve by completing the square.\newlinem2−10m−29=0m^2 - 10m - 29 = 0\newlineWrite your answers as integers, proper or improper fractions in simplest form, or decimals rounded to the nearest hundredth.\newline`m` = ____ or `m` = _____
Get tutor helpright-arrow

Posted 8 months ago

Question
Find g(x) g(x) , where g(x) g(x) is the translation 5 5 units up of f(x)=x2 f(x)=x^2 .\newlineWrite your answer in the form a(x–h)2+k a(x–h)^2+k , where a a , h h , and k k are integers.\newlineg(x)= g(x)= ____
Get tutor helpright-arrow

Posted 7 months ago

Question
What is the range of this quadratic function?\newliney=x2−4x+4y = x^2 - 4x + 4\newlineChoices:\newline{y∣y≥2}\left\{y \mid y \geq 2\right\}\newline{y∣y≤0}\left\{y \mid y \leq 0\right\}\newline{y∣y≥0}\left\{y \mid y \geq 0\right\}\newlineall real numbers\text{all real numbers}
Get tutor helpright-arrow

Posted 5 months ago

Question
Write the equation of the parabola that passes through the points (1,0)(1,0), (2,0)(2,0), and (3,–16)(3,\text{–}16). Write your answer in the form y=a(x–p)(x–q)y = a(x – p)(x – q), where aa, pp, and qq are integers, decimals, or simplified fractions.\newline______
Get tutor helpright-arrow

Posted 5 months ago

Question
Complete the square. Fill in the number that makes the polynomial a perfect-square quadratic.\newlinef2+8f+_____f^2 + 8f + \_\_\_\_\_
Get tutor helpright-arrow

Posted 5 months ago

Question
Solve for h h .\newlineh2+39h=0 h^2 + 39h = 0 \newline\newlineWrite each solution as an integer, proper fraction, or improper fraction in simplest form. If there are multiple solutions, separate them with commas.\newlineh= h = ____\newline
Get tutor helpright-arrow

Posted 5 months ago

Question
Write a quadratic function with zeros −9-9 and −7-7.\newlineWrite your answer using the variable xx and in standard form with a leading coefficient of 11.\newlinef(x)=_____f(x) = \_\_\_\_\_
Get tutor helpright-arrow

Posted 5 months ago

Question
Find the equation of the axis of symmetry for the parabola y=x2y = x^2. \newlineSimplify any numbers and write them as proper fractions, improper fractions, or integers.\newline‾\underline{\hspace{3cm}}
Get tutor helpright-arrow

Posted 4 months ago

Question
Find g(x)g(x), where g(x)g(x) is the translation 88 units up of f(x)=x2f(x) = x^2.\newlineWrite your answer in the form a(x–h)2+ka(x – h)^2 + k, where aa, hh, and kk are integers.\newlineg(x)=g(x) = ______\newline
Get tutor helpright-arrow

Posted 4 months ago

Question
Solve for xx.\newlinex2=1x^2 = 1\newline\newlineWrite your answer in simplified, rationalized form.\newlinex=x = ______ or x=x = ______\newline
Get tutor helpright-arrow

Posted 5 months ago

View all (96)

Related topics

Algebra - Order of OperationsAlgebra - Distributive Property`X` and `Y` AxesGeometry - Scalene TriangleCommon MultipleGeometry - Quadrant