Evaluate the derivative of the following functions.
f(x)...

Question

Evaluate the derivative of the following functions.

f(x) = x² + 2x³ cot^-1 x - ln (1 + x²)

Accepted Answer

Welcome back, everyone. Find the derivative of F of X equals 5x the power of 4 + 3x2 inverts of tangent of X plus LN X2 + 9. So let's write down the expression of the derivative. We have F of X. This is what we want to evaluate. For the first term we can factor out the 5, and then we are evaluating the derivative of X to the power of 4th. Then for the second term we have a product of two functions, so we're going to factor out the 3 and then differentiate X squared multiplied by the inverse of tangent of X. And then for the final term we want to evaluate the derivative of the natural logarithm. Of X2 + 9. So let's focus on H derivative. The first one that we want to evaluate is X to the power of 4th, and we have to recall that we can differentiate this using the power rule. So we bring down the exponent, the derivative of X to the power of N is equal to N, multiplied by X to the power of N minus 1. We bring down the 4 and then we decrease the original exponent by 1 unit. So 4 minus 1 gives us 3, and we have our first derivative. Now for the second derivative we have x2 multiplied by the inverse of 10 of X and here we want to apply the product rule. Let's recall that if we have a product g multiplied by h. And we want to validate the derivative of this product. We take the derivative of G multiplied by H, and we add the G multiplied by H derivative. So in this case, G is X2, so G is 2 X based on the power rule, and then our H. is inverse of 10 and each prime based on the table of basic derivatives is 1 divided by 1 plus x 2. That's the derivative of inverse of tangent. So essentially in this case we take the derivative of X2, which is 2 X, multiply that by h which is inverse of tangent of X. And then we add G, which is x2 multiplied by H which is 1 divided by 1 + X2, and that's our second result. Now for the final one we want to evaluate the derivative of LN of X2 + 9, and we have to recall that the derivative of LN of X is 1 divided by X. So in this case we would have 1 divided by X2 + 9 because this is our angle. But because our angle is not X and it is a function according to the chain rule, we have to multiply that by the derivative of X2 + 9, and here we can apply the power rule once again. We're going to get 2 x + 0 because the derivative of 9 is 0 and we get 2 X divided by X2 + 9, and that's our third result. And now combining 12, and 3, we can say that that the derivative of F of X is equal to now 5 multiplied by the result from the first step. So 5 multiplied by 4 gives us 20 and we have X cubed. Then we have 3 multiplied by the result from the second step. So 3 multiplied by 2 gives us 6. We're going to write 6 X. Inverse of tangent of X plus. We multiplied by x2 multiplied by 1 gives us we x2 divided by 1 + X2, and then we are adding the result from the third step, which is 2 x divided by x2 + 9. And this gives us our final answer. So let's label it and thank you for watching.

Evaluate the derivative of the following functions.
f(x) = x² + 2x³ cot^-1 x - ln (1 + x²)

Key Concepts

Derivative

Product and Chain Rules

Logarithmic Differentiation

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