# How do you differentiate #f(x)=(x+2)^2(x-5)^3# using the product rule?

# f'(x) = (x-5)^2(x+2)(5x - 4) #

for a function f)x) = g(x).h(x) ie. a product of 2 functions

then f'(x) = g(x).h'(x) + h(x).g'(x).............................(A)

substituting back into ( A) gives :

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To differentiate ( f(x) = (x+2)^2(x-5)^3 ) using the product rule, follow these steps:

- Identify the two functions being multiplied: ( u(x) = (x+2)^2 ) and ( v(x) = (x-5)^3 ).
- Apply the product rule: ( f'(x) = u'(x)v(x) + u(x)v'(x) ).
- Find the derivatives of ( u(x) ) and ( v(x) ):
- ( u'(x) = 2(x+2)(1) ) (using the power rule and chain rule)
- ( v'(x) = 3(x-5)^2 ) (using the power rule and chain rule)

- Substitute the derivatives into the product rule formula: ( f'(x) = 2(x+2)(x-5)^3 + (x+2)^2(3(x-5)^2) ).
- Simplify the expression if necessary.

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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.

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