Let ƒ(x) = (x - 3) (x + 3)²

a. Verify that ƒ'...

Question

Let ƒ(x) = (x - 3) (x + 3)²

a. Verify that ƒ'(x) = 3(x - 1) (x + 3) and ƒ"(x) = 6 (x + 1).

Accepted Answer

Welcome back, everyone. Consider the function H X equals x minus 5 multiplied by x + 22. Evaluate the first and the second derivatives of H X. Let's begin with the first derivative H X. And here we have a product of two functions X minus 5 multiplied by x + 22. So according to the product rule, and we have to take the derivative of the first function x minus 5 derivative. Multiply that by the 2nd function which is x + 2 squad. And then we have to add the value of the first function x minus 5 multiplied by the derivative of the second function, so X plus 2 squad derivative. And that's it. Now let's apply the power rule for the first term X minus 5. It gives us 1 minus 0 because the derivative of X is 1 and the derivative of -5 is 0 because -5 is a constant. So we simply get X + 2 squared. And then we are adding X minus 5. Multiplied by a composite function. So here we have to apply the chain rule. We have an exponential function. So let's apply the power rule. We bring down the two. We decrease the original exponent by 1 unit, which means that we end up with X + 2 to the power of 1, and then based on the chain rule, we have to multiply by the derivative of the inner function, so the derivative of X + 2. Now the derivative of X + 2 is equal to 1 because the derivative of X is 1 and the derivative of 2 is 0. So we can simply combine the terms. We have X + 2 squared. Plus 2 x + 2 multiplied by x minus 5, and we can factor out the common factor which is X + 2. Which gives us X + 2 + 2. X minus 5 for the remaining factor. Overall we can simplify S X + 2. Multiplied by X + 2 X, this gives us 3 X, and then for the other like terms, we have 2 + 2 multiplied by -5, so 2 minus 10. Overall this gives us -8, and this is how we get our first derivative of X, so we can label it as our first answer and continue with the second derivative which is H of X, and we can identify it using the first derivative. So let's differentiate the previous result. Once again, according to the power rule, we're going to take the derivative of the first function x + 2, multiply by 3x minus 8 the second function, and we're going to add the first function x + 2 multiplied by the derivative of 3x minus 8. So once again we're applying the power rule. We already know that the derivative of X + 2 is equal to 1. And now for the derivative of 3 X minus 8, we can factor out the constant. It gives us 3. The derivative of X is 1, and the derivative of -8 is 0. So overall, the derivative is 3. So we're going to get 3 X minus 8. Plus 3 in X + 2. Now let's simplify. E X plus 3X gives us 6X. We're distributing the terms and combining the like terms, and then we have negative 8 + 3 multiplied by 2. So -8 + 6. Gives us 2. And now if we factor. We can say that this is 2 in 3 x minus 1, which is H of X, the second derivative. And now we have the required answers. So let's label the 2nd 1 and conclude the video. Thank you for watching.

Let ƒ(x) = (x - 3) (x + 3)²a. Verify that ƒ'(x) = 3(x - 1) (x + 3) and ƒ"(x) = 6 (x + 1).

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Let ƒ(x) = (x - 3) (x + 3)²

a. Verify that ƒ'(x) = 3(x - 1) (x + 3) and ƒ"(x) = 6 (x + 1).