Differentiation from first principles- - cant do at all

pat666

Homework Statement

see attachment- have to do this because i can't figure out how to do the notation in this part sorry... I have no idea where to go with this and probably need quite a bit of help with it--- thanks.

The Attempt at a Solution

Attachments

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Homework Helper
The problem says "use the definition of derivative
$$\lim\frac{\delta y}{\delta x}$$

Do you know what that means?
$$\frac{\delta y}{\delta x}= \frac{y(1+ \delta x)- y(1)}{\delta x}$$

$y(1)= 1^2- 1= 0$ and $y(1+ \delta x)= (1+ \delta x)^2- (1+ \delta x)$.

pat666
hey - sorry bud i don't get any of that? - the lecture that we had talked for about 15s on this and I really don't understand it. more help would be GREATLY appreciated.

Mentor
The derivative definition is usually presented using upper-case delta, $\Delta$ rather than lower-case delta, $\delta$ as you have.

It might be helpful to use function notation, letting f(x) = y = x2 - x. The derivative of f at 1 can be written this way:
$$\lim_{\Delta x \to 0} \frac{f(1 + \Delta x) - f(1)}{\Delta x}$$

The fraction gives the slope of a secant line between (1, f(1)) and (1 + $\Delta x$, f(1 + $\Delta x$)). The numerator gives the vertical change (rise) and the denominator gives the horizontal change (run). As $\Delta x$ approaches zero, the slope of the secant line approaches the slope of the tangent line.

Substitute for f(1) and f(1 + $\Delta x$) in the limit formula above, simplify, and then take the limit.

If you still don't understand, your text should have an explanation of this and some examples.

Mentor
BTW, you should post calculus problems (like this one) in the Calculus & Beyond section.