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How do you use the chain rule to differentiate #y=x^3(2x-5)^4#?

Answer 1

Isaiah Allen

#dy/dx = x^2(2x-5)^3(14x-15)#

#y =x^3(2x-5)^4#

#dy/dx= x^3 * d/dx(2x-5)^4 + (2x-5)^4 * d/dx(x^3)# (Product rule)

#= x^3 * 4(2x-5)^3 * 2 + (2x-5)^4 * 3x^2# (Power rule and Chain rule)

#= x^2(2x-5)^3(8x+3(2x-5))#

#=x^2(2x-5)^3(14x-15)#

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Answer 2

Paisley Johnson

To differentiate ( y = x^3(2x - 5)^4 ) using the chain rule, follow these steps:

Identify the outer function ( u = x^3 ) and the inner function ( v = (2x - 5)^4 ).
Compute the derivative of the outer function ( \frac{du}{dx} = 3x^2 ).
Compute the derivative of the inner function ( \frac{dv}{dx} = 4(2x - 5)^3 ).
Apply the chain rule formula ( \frac{dy}{dx} = \frac{du}{dx} \cdot v + u \cdot \frac{dv}{dx} ).
Substitute the derivatives and functions into the chain rule formula.

So, the derivative of ( y = x^3(2x - 5)^4 ) is:

[ \frac{dy}{dx} = 3x^2 \cdot (2x - 5)^4 + x^3 \cdot 4(2x - 5)^3 ]

Simplify this expression if necessary.

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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.