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Use the Chain Rule lo prove the following.

(a) The derivative of an even (unction is an odd function.

(b) The derivative of an odd (unction is an even function.

a. $f^{\prime}(-x)=-f^{\prime}(x)$

b. $f^{\prime}(-x)$

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Oregon State University

University of Nottingham

Idaho State University

Okay, we're going to start this problem by approving that The derivative of uneven function is an odd function. So remember this relationship for even functions opposite X values have the same y value, and we can state that as f of X equals f of the opposite of X. So if the function is even, I should be able to take the derivative of the left side, and that should be equal to the derivative of the right side. Okay, so then let's work on the derivative of the right side. So we see that we have a composite function. The outer function is F of X on the inside. Function is the opposite of X. So we're going to have the derivative of the outer function first and then multiplied by the derivative of the inter function on the inner function is negative. X. So it's derivative would be negative one. Now, if I clean that up a little bit and just move the extra negative sign to the front, I have the opposite of F prime of the opposite of X. So if you take a look at that that fits our description of odd functions, odd functions have this relationship f of X equals the opposite of F of the opposite of X, and we see that that is what's holding true here for the derivative. Now we're going to prove that the derivative of an odd function is an even function. So here's the relationship that holds true for odd functions. So we should be able to take the derivative of both sides and have those be equal. Okay, so then when I take the derivative of the right side, I'm going to leave the constant negative one. I'm going to multiply by the derivative of the outside first, that would be f prime of the opposite of X and then multiply by the derivative of the inside. And that would be negative one. And we can multiply these negatives together and we get a positive. So get f prime of the Office of X. So we have f Prime of X is equal to the two f prime of the opposite of X. And remember what we said a minute ago for even functions they hold this relationship f of X equals f of the opposite of X. So that's what we found to be true for these derivatives.

Oregon State University