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redshift
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I need to differentiate z = artan(y/x) with respect to x. Somehow z' = y/(1 +(1/x)^2) doesn't seem right.
redshift said:I need to differentiate z = artan(y/x) with respect to x. Somehow z' = y/(1 +(1/x)^2) doesn't seem right.
HallsofIvy said:[tex] z'= \frac{1}{1+(\frac{y}{x})^2} \(-\frac{y}{x^2}\)[/tex]
by the chain rule.
Now multiply both numerator and denominator by x2.
Differentiation is a mathematical process of finding the rate at which a quantity changes with respect to another quantity. In simpler terms, it is the process of finding the slope of a curve at a specific point.
Differentiation is important because it allows us to analyze and understand the behavior of functions, such as finding maximum and minimum points, determining concavity, and solving optimization problems. It also has many practical applications in science, engineering, and economics.
To differentiate a function, you need to use the rules of differentiation, such as the power rule, product rule, quotient rule, and chain rule. These rules allow you to find the derivative of a function, which represents the rate of change of that function.
A 'simple' differentiation problem is one that involves finding the derivative of a basic function, such as a polynomial, exponential, or trigonometric function. These problems typically do not require the use of advanced techniques or multiple rules of differentiation.
Some common mistakes to avoid in differentiation include forgetting to use the chain rule, not simplifying the final answer, and making algebraic errors. It is also important to carefully check the domain of the original function and the derivative to ensure that the derivative is defined at the point of interest.