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How do you use the Product Rule to find the derivative of ##?

Answer 1

Samantha Adkison

See the explanation.

#h(x)=f(x)*g(x)#

#h'(x)=f'(x)*g(x)+f(x)*g'(x)#

Example:

#h=(x^4-5x^2+1)e^x#

#f=x^4-5x^2+1 => f'=4x^3-10x# #g=e^x => g'=e^x#

#h'=(4x^3-10x)e^x+(x^4-5x^2+1)e^x#

#h'=(x^4+4x^3-5x^2-10x+1)e^x#

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Answer 2

Axel Alvarez

To use the Product Rule to find the derivative of a function, follow these steps:

Identify the two functions being multiplied together.
Differentiate each function separately.
Apply the Product Rule formula, which states that the derivative of the product of two functions u(x) and v(x) is given by: [ (u(x) \cdot v(x))' = u'(x) \cdot v(x) + u(x) \cdot v'(x) ]

Where:

( u'(x) ) is the derivative of the first function.
( v'(x) ) is the derivative of the second function.
( u(x) ) and ( v(x) ) are the original functions.

Substitute the derivatives and original functions into the formula.
Simplify the resulting expression if possible.

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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.