Let's look at your difference quotient in LaTeX.jmanna98 said:The first thing I did was sub the function into the derivative formula which made a huge mess of a problem to simplify.
[(-1/sqrt2(x+deltax)) +2(x+deltax)] - [(-1/sqrt2x)+2x] all over deltax.
I am a little rusty on working with radicals and tried a few things that ended up in a mess but I am thinking that I should simplify the numerator of this first by finding the LCD of the rational expressions in the numberator of the whole problem. LCD:(sqrt2(x+deltax))(sqrt2x)? Then multiply by the conjugate? I sort of feel on the right track but at the same time I feel that my LCD is incorrect for some reason.
A derivative is a mathematical concept that represents the rate of change of a function at a specific point. It is the slope of the tangent line to the function at that point.
A rational function is a function that can be expressed as the ratio of two polynomials. It can also be written in the form of f(x) = p(x)/q(x), where p(x) and q(x) are polynomials.
A radical function is a function that contains a radical expression, such as a square root, cube root, or higher root. It can also be written in the form of f(x) = √(x) or f(x) = ∛(x).
To find the derivative of a rational function, you can use the quotient rule, which states that the derivative of f(x)/g(x) is equal to (f'(x)g(x) - f(x)g'(x))/[g(x)]^2. Alternatively, you can rewrite the function as f(x) = p(x)q(x)^-1 and use the power rule to find the derivative.
To find the derivative of a radical function, you can use the chain rule, which states that the derivative of √(f(x)) is equal to 1/2√(f(x)) * f'(x). Similarly, for ∛(f(x)), the derivative would be 1/3∛(f(x)) * f'(x). You can also rewrite the function using rational exponents and use the power rule to find the derivative.