How do you use the chain rule to differentiate #y=(4x^3-7)^4(3x+2)^10#?
I got:
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:
Chain Rule:
Therefore, what you have (without the chain rule) is:
Now, incorporate the chain rule to finish this step and get:
Now, group terms before factoring:
Factor out common terms:
Factor a little bit more:
And maybe distribute inner terms:
That's probably simplified enough. Either way, it's correct.
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To differentiate ( y = (4x^3 - 7)^4(3x + 2)^{10} ) using the chain rule, follow these steps:
- Identify the functions within the expression: ( u = 4x^3 - 7 ) and ( v = 3x + 2 ).
- Compute the derivatives of ( u ) and ( v ): ( \frac{du}{dx} ) and ( \frac{dv}{dx} ).
- Apply the chain rule: ( \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} + \frac{dy}{dv} \cdot \frac{dv}{dx} ).
- Calculate ( \frac{dy}{du} ) and ( \frac{dy}{dv} ).
- 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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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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