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:

2x^(2)(x^(3)-x)-3x(x^(4)+2x)-2(x^(4)-3x^(2))

2x2(x3−x)−3x(x4+2x)−2(x4−3x2) 2 x^{2}\left(x^{3}-x\right)-3 x\left(x^{4}+2 x\right)-2\left(x^{4}-3 x^{2}\right)

View step-by-step helpView step-by-step help
Homeright chevron icon
Math Problemsright chevron icon
Calculusright chevron icon
Find derivatives of using multiple formulaeright chevron icon

Full solution

Q. 2x2(x3−x)−3x(x4+2x)−2(x4−3x2) 2 x^{2}\left(x^{3}-x\right)-3 x\left(x^{4}+2 x\right)-2\left(x^{4}-3 x^{2}\right)
  1. Find Derivative: We need to find the derivative of the given function with respect to xx. The function is a polynomial, and we will use the power rule, product rule, and the sum rule for differentiation.
  2. Apply Product Rule: First, let's apply the product rule to the first term 2x2(x3−x)2x^{2}(x^{3}-x). The product rule states that (d/dx)(u∗v)=u′v+uv′(d/dx)(u*v) = u'v + uv', where uu and vv are functions of xx. Let u=2x2u = 2x^2 and v=x3−xv = x^3 - x.
  3. Simplify First Term: Now, we differentiate uu and vv with respect to xx.
    u′=ddx(2x2)=4xu' = \frac{d}{dx}(2x^2) = 4x
    v′=ddx(x3−x)=3x2−1v' = \frac{d}{dx}(x^3 - x) = 3x^2 - 1
  4. Apply Product Rule: Applying the product rule, we get the derivative of the first term:\newline(2x2)′(x3−x)+2x2(x3−x)′=4x(x3−x)+2x2(3x2−1)(2x^2)'(x^3 - x) + 2x^2(x^3 - x)' = 4x(x^3 - x) + 2x^2(3x^2 - 1)
  5. Simplify Second Term: Now, let's simplify the expression we just found:\newline4x(x3−x)+2x2(3x2−1)=4x4−4x2+6x4−2x24x(x^3 - x) + 2x^2(3x^2 - 1) = 4x^4 - 4x^2 + 6x^4 - 2x^2\newlineCombining like terms, we get:\newline10x4−6x210x^4 - 6x^2
  6. Apply Power Rule: Next, we apply the product rule to the second term −3x(x4+2x)-3x(x^4 + 2x). Let u=−3xu = -3x and v=x4+2xv = x^4 + 2x.
  7. Combine Derivatives: Differentiate uu and vv with respect to xx.u′=ddx(−3x)=−3u' = \frac{d}{dx}(-3x) = -3v′=ddx(x4+2x)=4x3+2v' = \frac{d}{dx}(x^4 + 2x) = 4x^3 + 2
  8. Simplify Expression: Applying the product rule, we get the derivative of the second term:\newline(−3x)′(x4+2x)+(−3x)(x4+2x)′=−3(x4+2x)+(−3x)(4x3+2) (-3x)'(x^4 + 2x) + (-3x)(x^4 + 2x)' = -3(x^4 + 2x) + (-3x)(4x^3 + 2)
  9. Simplify Expression: Applying the product rule, we get the derivative of the second term:\newline(−3x)′(x4+2x)+(−3x)(x4+2x)′=−3(x4+2x)+(−3x)(4x3+2)(-3x)'(x^4 + 2x) + (-3x)(x^4 + 2x)' = -3(x^4 + 2x) + (-3x)(4x^3 + 2)Now, let's simplify the expression we just found:\newline−3(x4+2x)+(−3x)(4x3+2)=−3x4−6x−12x4−6x-3(x^4 + 2x) + (-3x)(4x^3 + 2) = -3x^4 - 6x - 12x^4 - 6x\newlineCombining like terms, we get:\newline−15x4−12x-15x^4 - 12x
  10. Simplify Expression: Applying the product rule, we get the derivative of the second term:\newline(−3x)′(x4+2x)+(−3x)(x4+2x)′=−3(x4+2x)+(−3x)(4x3+2)(-3x)'(x^4 + 2x) + (-3x)(x^4 + 2x)' = -3(x^4 + 2x) + (-3x)(4x^3 + 2)Now, let's simplify the expression we just found:\newline−3(x4+2x)+(−3x)(4x3+2)=−3x4−6x−12x4−6x-3(x^4 + 2x) + (-3x)(4x^3 + 2) = -3x^4 - 6x - 12x^4 - 6x\newlineCombining like terms, we get:\newline−15x4−12x-15x^4 - 12xFor the third term −2(x4−3x2)-2(x^4 - 3x^2), we can directly apply the power rule since it's a simple polynomial.\newline(d/dx)(−2x4+6x2)=−8x3+12x(d/dx)(-2x^4 + 6x^2) = -8x^3 + 12x
  11. Simplify Expression: Applying the product rule, we get the derivative of the second term:\newline(−3x)′(x4+2x)+(−3x)(x4+2x)′=−3(x4+2x)+(−3x)(4x3+2)(-3x)'(x^4 + 2x) + (-3x)(x^4 + 2x)' = -3(x^4 + 2x) + (-3x)(4x^3 + 2)Now, let's simplify the expression we just found:\newline−3(x4+2x)+(−3x)(4x3+2)=−3x4−6x−12x4−6x-3(x^4 + 2x) + (-3x)(4x^3 + 2) = -3x^4 - 6x - 12x^4 - 6x\newlineCombining like terms, we get:\newline−15x4−12x-15x^4 - 12xFor the third term −2(x4−3x2)-2(x^4 - 3x^2), we can directly apply the power rule since it's a simple polynomial.\newline(d/dx)(−2x4+6x2)=−8x3+12x(d/dx)(-2x^4 + 6x^2) = -8x^3 + 12xNow, we combine the derivatives of all three terms to get the derivative of the entire function:\newline10x4−6x2−15x4−12x−8x3+12x10x^4 - 6x^2 - 15x^4 - 12x - 8x^3 + 12x
  12. Simplify Expression: Applying the product rule, we get the derivative of the second term:\newline(−3x)′(x4+2x)+(−3x)(x4+2x)′=−3(x4+2x)+(−3x)(4x3+2)(-3x)'(x^4 + 2x) + (-3x)(x^4 + 2x)' = -3(x^4 + 2x) + (-3x)(4x^3 + 2)Now, let's simplify the expression we just found:\newline−3(x4+2x)+(−3x)(4x3+2)=−3x4−6x−12x4−6x-3(x^4 + 2x) + (-3x)(4x^3 + 2) = -3x^4 - 6x - 12x^4 - 6x\newlineCombining like terms, we get:\newline−15x4−12x-15x^4 - 12xFor the third term −2(x4−3x2)-2(x^4 - 3x^2), we can directly apply the power rule since it's a simple polynomial.\newline(d/dx)(−2x4+6x2)=−8x3+12x(d/dx)(-2x^4 + 6x^2) = -8x^3 + 12xNow, we combine the derivatives of all three terms to get the derivative of the entire function:\newline10x4−6x2−15x4−12x−8x3+12x10x^4 - 6x^2 - 15x^4 - 12x - 8x^3 + 12xSimplify the expression by combining like terms:\newline(10x4−15x4)−6x2−8x3+(−12x+12x)(10x^4 - 15x^4) - 6x^2 - 8x^3 + (-12x + 12x)\newlineThis simplifies to:\newline−5x4−8x3−6x2-5x^4 - 8x^3 - 6x^2

More problems from Find derivatives of using multiple formulae

Question
Find the derivative of f(x) f(x) .\newlinef(x)=cos⁡(x) f(x) = \cos(x) \newlinef′(x)= f'(x) = ______
Get tutor helpright-arrow

Posted 7 months ago

Question
Find the derivative of f(x) f(x) .\newlinef(x)=tan⁡(x) f(x) = \tan(x) \newlinef′(x)= f'(x) = ______
Get tutor helpright-arrow

Posted 8 months ago

Question
Find the derivative of f(x) f(x) .\newlinef(x)=ex f(x) = e^x \newlinef′(x)= f'\left(x\right) = ______
Get tutor helpright-arrow

Posted 8 months ago

Question
Find the derivative of f(x) f(x) .\newlinef(x)=ln⁡(x) f(x) = \ln(x) \newlinef′(x)= f'(x) = ______
Get tutor helpright-arrow

Posted 8 months ago

Question
Find the derivative of f(x) f(x) .\newlinef(x)=tan⁡−1x f(x) = \tan^{-1}{x} \newlinef′(x)= f'(x) = ______
Get tutor helpright-arrow

Posted 8 months ago

Question
Find the derivative of f(x) f(x) .\newlinef(x)=xex f(x) = x e^x \newlinef′(x)= f'\left(x\right) = ______
Get tutor helpright-arrow

Posted 8 months ago

Question
Find the derivative of f(x) f(x) . \newlinef(x)=exx f(x) = \frac{e^x}{x} \newlinef′(x)= f'(x) = ______
Get tutor helpright-arrow

Posted 8 months ago

Question
Find the derivative of f(x) f(x) . \newline f(x)=e(x+1) f(x) = e^{(x + 1)} \newline f′(x)= f'\left(x\right) = ______
Get tutor helpright-arrow

Posted 7 months ago

Question
Find the derivative of f(x) f(x) . \newlinef(x)=x+3 f(x) = \sqrt{x+3} \newlinef′(x)= f'(x) = ______
Get tutor helpright-arrow

Posted 8 months ago

Question
Find the derivative of g(x) g(x). \newline g(x)=ln⁡(2x) g(x) = \ln(2x) \newline g′(x)= g^{\prime}(x) = ______
Get tutor helpright-arrow

Posted 9 months ago

View all (358)

Related topics

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