- #1
tahayassen
- 270
- 1
Homework Statement
f(x) = 2x * (10-x^2)^(1/2)
Determine f'(x).
2. The attempt at a solution
f'(x) = 2x*(1/2) * (10-x^2)^(-1/2) * (-2x)
= x * (-2x) * (10-x^2)^(-1/2)
= -2x^2 * (10-x^2)^(-1/2)
What am I doing wrong?
HallsofIvy said:You aren't using the product rule. You have multiplied x by the derivative of [itex](10- x^2)^{1/2}[/itex], effectively treating that first "x" as though it were a constant. It is not.
A function is a mathematical relationship between two quantities, where for each input there is a unique output. It can also be described as a rule or set of instructions that maps an input to an output.
To differentiate a function, you need to find its derivative. This can be done by taking the limit of the difference quotient as the change in input approaches zero. Alternatively, you can use differentiation rules and formulas to find the derivative of the function.
Differentiating a function helps us to understand how the function changes with respect to its input. This can be used to find the rate of change, extrema, and concavity of the function, which can be useful in applications such as optimization and curve sketching.
Some common differentiation rules include the power rule, product rule, quotient rule, and chain rule. These rules help us to find the derivative of more complicated functions by breaking them down into simpler parts.
No, not all functions can be differentiated. Functions that are not continuous or do not have a defined slope at a certain point cannot be differentiated. Additionally, some functions may be too complex to differentiate using traditional methods, but may still have a derivative using more advanced techniques.