The derivative of 6x/(x^2+3)^2 is (18x^2-36x+18)/(x^2+3)^3.
To solve the derivative of 6x/(x^2+3)^2, we can use the quotient rule. First, we find the derivative of the numerator, which is 6. Then, we find the derivative of the denominator, which is 2(x^2+3) multiplied by the derivative of x^2+3, which is 2x. We then put these values into the quotient rule formula, (f'g-g'f)/g^2, where f' represents the derivative of the numerator and g' represents the derivative of the denominator. Simplifying the formula gives us (18x^2-36x+18)/(x^2+3)^3 as the final answer.
The purpose of finding the derivative of a function is to determine the rate of change of that function at a specific point. In this case, the derivative of 6x/(x^2+3)^2 tells us the rate of change of the original function at a specific x-value. It can also be used to find the slope of a tangent line to the graph of the function at that point.
Yes, the derivative of 6x/(x^2+3)^2 can be simplified further by factoring out a common factor of 6 from the numerator, giving us 6(3x^2-6x+3)/(x^2+3)^3. However, this is considered to be in its simplest form as it cannot be further reduced without changing the meaning of the original function.
Yes, there are many real-life applications of finding the derivative of a function. For example, in physics, the derivative of a position function can be used to find the velocity of an object at a specific time. In economics, the derivative of a cost function can be used to find the marginal cost of producing an additional unit of a product. In engineering, the derivative of a displacement function can be used to find the acceleration of a moving object. The list goes on, as the concept of rate of change is applicable to many fields of study.