How do you use the chain rule to differentiate #y=(2x^3+7)^6(2x+1)^8#?

Answer 1

#36x^2(2x^3+7)^5(2x+1)^8+16(2x+1)^7(2x^3+7)^6#

The first step here is to recognize that we are first going to use the product rule followed by the chain rule when we take the derivative:

#f'(x)g(x)+g'(x)f(x)#
It doesn't matter who is #f(x)# or who is #g(x)#
I'll let #f(x)# be the first function #(2x^3+7)^6#
And #g(x)# be the second function #(2x+1)^8#

Following the product rule, lets do the first half:

#f'(x) times g(x)#
#f'(x) times g(x)=6(2x^3+7)^5(6x^2)xx(2x+1)^8#
The #6x^2# came from the chain rule, where you take the derivative of the outside and then multiply it by derivative of the inside. Notice that the derivative of the inside function #(2x^3+7)# is #6x^2# Our #g(x)# remains untouched. We don't need the #xx# sign I left it there to illustrate the point.

Now lets do the other half of product rule:

#g'(x)f(x)#
#g'(x)f(x)=8(2x+1)^7(2)xx(2x^3+7)^6#
The derivative of the inside function #(2x+1)# is just #2#

Now just put them to together:

#6(2x^3+7)^5(6x^2)(2x+1)^8+8(2x+1)^7(2)(2x^3+7)^6#

You can simplify it:

#36x^2(2x^3+7)^5(2x+1)^8+16(2x+1)^7(2x^3+7)^6#
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Answer 2

To differentiate ( y = (2x^3 + 7)^6(2x + 1)^8 ) using the chain rule, you first find the derivative of the outer function multiplied by the derivative of the inner function, then multiply them together.

Let ( u = 2x^3 + 7 ) and ( v = 2x + 1 ).

( \frac{dy}{dx} = \frac{d}{dx}[(2x^3 + 7)^6(2x + 1)^8] )

Apply the chain rule:

( \frac{dy}{dx} = \frac{d}{du}(u^6) \cdot \frac{du}{dx} \cdot v^8 + u^6 \cdot \frac{d}{dv}(v^8) \cdot \frac{dv}{dx} )

Differentiate each part:

( \frac{dy}{du} = 6u^5 )

( \frac{du}{dx} = 6x^2 )

( \frac{dy}{dv} = 8v^7 )

( \frac{dv}{dx} = 2 )

Plug the values back in:

( \frac{dy}{dx} = (6(2x^3 + 7)^5) \cdot (6x^2) \cdot (2x + 1)^8 + (2x^3 + 7)^6 \cdot (8(2x + 1)^7) \cdot 2 )

( \frac{dy}{dx} = 12(2x^3 + 7)^5 \cdot 6x^2 \cdot (2x + 1)^8 + 2(2x^3 + 7)^6 \cdot 8(2x + 1)^7 )

Simplify the expression.

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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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