Derivative of xcos(logx) | Product and Chain Rule Method

Thread starter karnten07
Start date Dec 3, 2007
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Derivative

In summary, the product rule is a formula used to find the derivative of a product of two functions. To apply it, we use the formula f'(x)g(x) + f(x)g'(x). Similarly, the chain rule is a formula used to find the derivative of a composite function, and we use the formula f'(g(x))*g'(x) to apply it. Both rules can be used together to find the derivative of functions that involve both multiplication and composition, as seen in the example of xcos(logx).

Dec 3, 2007

karnten07

Homework Statement

I want to work out the derivative of xcos(log(x))

Homework Equations

The Attempt at a Solution

By using the product rule: and the chain rule;

cos(logx) - xsin(logx).1/x = cos(log(x)) - sin(log(x))

Is this right?

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Dec 3, 2007

buzzmath

looks right

Dec 3, 2007

karnten07

buzzmath said:

looks right

Thankyou

Related to Derivative of xcos(logx) | Product and Chain Rule Method

1. What is the product rule?

The product rule is a formula used to find the derivative of a function that is the product of two other functions. It states that the derivative of a product of two functions, f(x) and g(x), is equal to f'(x)g(x) + f(x)g'(x).

2. How do you apply the product rule to find the derivative of xcos(logx)?

To apply the product rule, we first identify the two functions in the product, which in this case are x and cos(logx). Then, we use the formula f'(x)g(x) + f(x)g'(x) to find the derivative, which would be (xcos(logx))' = x(-sin(logx)) + cos(logx)(1/x).

3. What is the chain rule?

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).

4. How do you apply the chain rule to find the derivative of xcos(logx)?

To apply the chain rule, we first identify the inner and outer functions in the composite function, which in this case are cos(logx) and x, respectively. Then, we use the formula f'(g(x))*g'(x) to find the derivative, which would be (xcos(logx))' = (-sin(logx))*(1/x) = -sin(logx)/x.

5. Can the product and chain rule be used together?

Yes, the product and chain rule can be used together to find the derivative of functions that involve both multiplication and composition. In the case of xcos(logx), we can use both rules to find the derivative, with the final answer being (-sin(logx))*(1/x) + cos(logx)(1/x) = (-sin(logx) + cos(logx))/x.