How do you use the chain rule to differentiate #y=(4x^3-7)^4(3x+2)^10#?

Answer 1

I got:

#6(4x^3 - 7)^3(3x + 2)^9(44x^3 + 16x^2 - 35)#

If you can't validly modify your answer to match this one, you've mis-factored somewhere.

You would use the product rule on these terms, and the chain rule while using the product rule.

Product Rule:

#\mathbf(d/(dx)[f(x)g(x)] = f(x)g"'"(x) + g(x)f"'"(x))#

Chain Rule:

#\mathbf(d/(dx)[f(g(x))] = f"'"(g(x))*g"'"(x))#

Therefore, what you have (without the chain rule) is:

#d/(dx)[(4x^3 - 7)^4(3x + 2)^10]#
#= (4x^3 - 7)^4*10(3x + 2)^9 + (3x + 2)^10*4(4x^3 - 7)^3#

Now, incorporate the chain rule to finish this step and get:

#= (4x^3 - 7)^4*10(3x + 2)^9*3 + (3x + 2)^10*4(4x^3 - 7)^3*12x^2#

Now, group terms before factoring:

#= 30(4x^3 - 7)^4(3x + 2)^9 + 48x^2(4x^3 - 7)^3(3x + 2)^10#

Factor out common terms:

#= (4x^3 - 7)^3[30(4x^3 - 7)(3x + 2)^9 + 48x^2(3x + 2)^10]#
#= (4x^3 - 7)^3(3x + 2)^9[30(4x^3 - 7) + 48x^2(3x + 2)]#

Factor a little bit more:

#= 6(4x^3 - 7)^3(3x + 2)^9[5(4x^3 - 7) + 8x^2(3x + 2)]#

And maybe distribute inner terms:

#= 6(4x^3 - 7)^3(3x + 2)^9(20x^3 - 35 + 24x^3 + 16x^2)#
#= color(blue)(6(4x^3 - 7)^3(3x + 2)^9(44x^3 + 16x^2 - 35))#

That's probably simplified enough. Either way, it's correct.

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Answer 2

To differentiate ( y = (4x^3 - 7)^4(3x + 2)^{10} ) using the chain rule, follow these steps:

  1. Identify the functions within the expression: ( u = 4x^3 - 7 ) and ( v = 3x + 2 ).
  2. Compute the derivatives of ( u ) and ( v ): ( \frac{du}{dx} ) and ( \frac{dv}{dx} ).
  3. Apply the chain rule: ( \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} + \frac{dy}{dv} \cdot \frac{dv}{dx} ).
  4. Calculate ( \frac{dy}{du} ) and ( \frac{dy}{dv} ).
  5. Substitute the values obtained in steps 4 and 2 into the chain rule formula to find ( \frac{dy}{dx} ).

Here are the detailed calculations:

[ \frac{du}{dx} = 12x^2 ] [ \frac{dv}{dx} = 3 ] [ \frac{dy}{du} = 4(4x^3 - 7)^3 \cdot 12x^2 ] [ \frac{dy}{dv} = 10(3x + 2)^9 \cdot 3 ] [ \frac{dy}{dx} = 4(4x^3 - 7)^3 \cdot 12x^2 + 10(3x + 2)^9 \cdot 3 ]

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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