Find the derivative of the given function

  • #1
chwala
Gold Member
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Homework Statement
See attached( I want to attempt the problem using quotient and product rule).
Relevant Equations
Differentiation
1692280784218.png


1692280869222.png


Let's see how messy it gets...

##\dfrac{dy}{dx}=\dfrac{(1-10x)(\sqrt{x^2+2})5x^4 -(x^5)(-10)(\sqrt{x^2+2})-x^5(1-10x)\frac{1}{2}(x^2+2)^{-\frac{1}{2}}2x}{[(1-10x)(\sqrt{x^2+2})]^2}##

##\dfrac{dy}{dx}=\dfrac{5x^4(1-10x)(x^2+2)+(10x^5(x^2+2))-x^6(1-10x)}{\sqrt{x^2+2}}⋅\dfrac{1}{[(1-10x)(\sqrt{x^2+2})]^2}####\dfrac{dy}{dx}=\dfrac{5x^4(1-10x)(x^2+2)+(10x^5(x^2+2))-x^6(1-10x)}{\sqrt{x^2+2}}⋅\dfrac{1}{[(1-10x)^2(\sqrt{x^2+2})^2]}##

##\dfrac{dy}{dx}=\dfrac{5x^4(1-10x)(x^2+2)+(10x^5(x^2+2))-x^6(1-10x)}{[(1-10x)^2(\sqrt{x^2+2})^3]}##

##\dfrac{dy}{dx}=\dfrac{5x^4}{(1-10x)\sqrt{x^2+2}}+\dfrac{10x^5}{[(1-10x)^2\sqrt{x^2+2}}-\dfrac{x^6}{[(1-10x)\sqrt{x^2+2})^3]}##

Factoring out ##\dfrac{1}{(1-10x)\sqrt{x^2+2}}## will give the desired result.

Bingo!! :cool:
 
Last edited:
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  • #3
chwala said:
Let's see how messy it gets...

##\dfrac{dy}{dx}=\dfrac{(1-10x)(\sqrt{x^2+2})5x^4 -(x^5)(-10x)(\sqrt{x^2+2})-x^5(1-10x)\frac{1}{2}(x^2+2)^{-\frac{1}{2}}2x}{[(1-10x)(\sqrt{x^2+2})]^2}##

##\dfrac{dy}{dx}=\dfrac{5x^4(1-10x)(x^2+2)+(10x^6(x^2+2))-x^6(1-10x)}{\sqrt{x^2+2}}⋅\dfrac{1}{[(1-10x)(\sqrt{x^2+2})]^2}##

##\dfrac{dy}{dx}=\dfrac{5x^4(1-10x)(x^2+2)+(10x^6(x^2+2))-x^6(1-10x)}{\sqrt{x^2+2}}⋅\dfrac{1}{[(1-10x)^2(\sqrt{x^2+2})^2]}##

checking latex a minute
That's pretty messy. The logarithmic differentiation that was recommended seems to be a lot simpler.
 
  • #4
Mark44 said:
That's pretty messy. The logarithmic differentiation that was recommended seems to be a lot simpler.
True...just a little exercise for the brain... :cool:
 
  • #5
I'd just like to note that, in the proposed solution by taking logs before differentiating, one should first simplify [itex]\ln(x^5) = 5 \ln x[/itex] and [itex]\ln(\sqrt{x^2 + 2}) = \frac12\ln(x^2 + 2)[/itex] before taking the derivative, thereby saving an application of the chain rule.
 
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Likes PhDeezNutz and chwala

FAQ: Find the derivative of the given function

What is the derivative of a constant function?

The derivative of a constant function is always zero. If f(x) = c, where c is a constant, then f'(x) = 0.

How do you find the derivative of a polynomial function?

To find the derivative of a polynomial function, apply the power rule: If f(x) = ax^n, then f'(x) = n*ax^(n-1). Apply this rule term-by-term for each term in the polynomial.

What is the chain rule and how is it used to find derivatives?

The chain rule is used to find the derivative of a composite function. If you have a function g(x) inside another function f(g(x)), the derivative is found using f'(g(x)) * g'(x).

How do you find the derivative of a trigonometric function?

To find the derivative of a trigonometric function, use specific derivative rules for each function: for example, the derivative of sin(x) is cos(x), and the derivative of cos(x) is -sin(x).

What is the product rule and how do you apply it?

The product rule is used to find the derivative of the product of two functions. If you have two functions u(x) and v(x), then the derivative of their product is given by u'(x)v(x) + u(x)v'(x).

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