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.

Homework Equations

The Attempt at a Solution

 

[HallsofIvy]

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.

 

[Mark44]

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.

 

[Mark44]

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