The derivative of the function f(x) = x2m isa)even functionb)odd functionc)constant functiond)None of theseCorrect answer is option 'B'. Can you explain this answer?

Question

Answer

The derivative of the function f(x) = x^2m is an odd function.

Explanation:

To find the derivative of the function f(x), we can use the power rule of differentiation. According to the power rule, if we have a function f(x) = x^n, the derivative of f(x) is given by:

f'(x) = n * x^(n-1)

In this case, the function f(x) = x^2m can be written as f(x) = (x^2)^m. Applying the power rule, we can find the derivative as follows:

f'(x) = m * (x^2)^(m-1) * 2x

Simplifying further:

f'(x) = 2mx^(2m-1)

Odd Function:

An odd function is a function that satisfies the property f(-x) = -f(x) for all values of x in the domain of the function. In other words, if we replace x with -x in the function and negate the result, it should be equal to the function itself.

Let's substitute -x for x in the derivative we found:

f'(-x) = 2m(-x)^(2m-1)

Now let's negate the result:

-f'(x) = -2m(-x)^(2m-1)

If we simplify this expression, we get:

-f'(x) = 2m(-x)^(2m-1)

We can observe that -f'(x) = f'(-x), which means that the derivative of f(x) satisfies the property of an odd function.

Conclusion:

The derivative of the function f(x) = x^2m is an odd function.

Answer

Here f(x) = x^2m is an even function f'(x) = 2mx^2m-1 is an odd function.

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