find derivative using limit definition:
f(x) = x - sqrt(x)
so f'(x) = lim
h->0 [f(x+h) - f(x)]/h
but I keep trying to solve by multiplying by the conjugate but I can't figure it out..there's nothing that can be cancelled or anything and I can't get the derivative
Some studies suggest that kidney mass K in mammals (in kilograms) is related to body mass m (in kilograms) by the approximate formula K = .007m^(0.85). Calculate dK/dm at m = 68. Then calculate the derivative with respect to m of the relative kidney-to-mass ratio K/m at m = 68.
ahh sorry that was supposed to be a new question. i need help with both though hahah, so pleaseee help me! i know derivatives but not in these contexts.
To find the derivative of the function f(x) = x - sqrt(x) using the limit definition, you can follow these steps:
Step 1: Begin with the limit definition of the derivative: f'(x) = lim(h->0) [f(x+h) - f(x)]/h.
Step 2: Substitute f(x) = x - sqrt(x) into the equation: f'(x) = lim(h->0) [(x + h - sqrt(x + h)) - (x - sqrt(x))]/h.
Step 3: Expand the expression inside the limit: f'(x) = lim(h->0) [x + h - sqrt(x + h) - x + sqrt(x)]/h.
Step 4: Simplify the expression inside the limit: f'(x) = lim(h->0) [h - (sqrt(x + h) - sqrt(x))]/h.
Step 5: To continue solving, we need to rationalize the numerator. Multiply the numerator and denominator by the conjugate of the numerator: f'(x) = lim(h->0) [h - (sqrt(x + h) - sqrt(x))]/h * [sqrt(x + h) + sqrt(x)]/[sqrt(x + h) + sqrt(x)].
Step 6: Distribute and simplify the numerator: f'(x) = lim(h->0) [h(sqrt(x + h) + sqrt(x)) - (x + h - x)]/[h(sqrt(x + h) + sqrt(x))].
Step 7: Cancel out terms in the numerator: f'(x) = lim(h->0) [h(sqrt(x + h) + sqrt(x)) - h]/[h(sqrt(x + h) + sqrt(x))].
Step 8: Simplify the numerator: f'(x) = lim(h->0) [h(sqrt(x + h) + sqrt(x) - 1)]/[h(sqrt(x + h) + sqrt(x))].
Step 9: Cancel out h terms in the numerator and denominator: f'(x) = lim(h->0) [(sqrt(x + h) + sqrt(x) - 1)]/[sqrt(x + h) + sqrt(x)].
Step 10: Now that h is approaching 0, substitute h = 0 into the expression: f'(x) = (sqrt(x) + sqrt(x) - 1)/(sqrt(x) + sqrt(x)).
Step 11: Simplify the expression: f'(x) = (2sqrt(x) - 1)/(2sqrt(x)).
Therefore, the derivative of f(x) = x - sqrt(x) is f'(x) = (2sqrt(x) - 1)/(2sqrt(x)).