Combining product rule with chain rule

Thread starter ziddy83
Start date Nov 7, 2004
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Chain Chain rule Product Product rule

In summary, the product rule is used to find the derivative of a product of two functions, while the chain rule is used to find the derivative of a composite function. They can be combined by applying the product rule to the derivative of the outer function and the chain rule to the derivative of the inner function. Both rules can also be used together in higher order derivatives.

Nov 7, 2004

ziddy83

Hey what's up,
The problem is...find the derivative of:

y= x^2 sec(x-5)

My question is...Would I start off by using the product rule and combine the chain rule with it, with x-5 being the inner function and sec(u) being the outer function? thanks...

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Nov 7, 2004

Dr Transport

Science Advisor

Insights Author

Gold Member

That is correct, use the product rule then the chain-rule.

Nov 7, 2004

ziddy83

ok.. thanks a lot

1. What is the product rule for differentiation?

The product rule is a formula used to find the derivative of a product of two functions. It states that the derivative of the product of two functions f(x) and g(x) is equal to the first function (f(x)) times the derivative of the second function (g'(x)) plus the second function (g(x)) times the derivative of the first function (f'(x)). This can be written as (f(x) * g(x))' = f(x) * g'(x) + g(x) * f'(x).

2. What is the chain rule for differentiation?

The chain rule is a formula used to find the derivative of a composite function. It states that the derivative of a composite function f(g(x)) is equal to the derivative of the outer function (f'(g(x))) multiplied by the derivative of the inner function (g'(x)). This can be written as (f(g(x)))' = f'(g(x)) * g'(x).

3. How do we combine the product rule and chain rule?

In order to combine the product rule and chain rule, we apply the product rule to the derivative of the outer function and the chain rule to the derivative of the inner function. This can be written as (f(g(x)))' = f'(g(x)) * g'(x). We then substitute the derivatives of the inner and outer functions into the formula to find the final derivative.

4. When should we use the product rule vs the chain rule?

The product rule should be used when taking the derivative of a product of two functions, while the chain rule should be used when taking the derivative of a composite function. If the function contains both a product and a composite, both rules may need to be used in combination.

5. Can the product rule and chain rule be used together in higher order derivatives?

Yes, the product rule and chain rule can be used together in higher order derivatives. When finding the second or higher order derivative of a product or composite function, we apply the respective rule(s) multiple times until the desired derivative is found.