# Math Help - Need help to understand how the slope was derived in this problem

1. ## Need help to understand how the slope was derived in this problem

I am kind of struggling to understand how the slope is derived in the following problem:
"Find the equation of the tangent line to the curve y = x^2 through the point (7, 49)."

The answer is that the slope is 2x but I don't understand how they got there since there is only one set of points given.

Thank you for the help,

Alex

2. $y'=\mbox{exponent}*x^{\mbox{exponent -1}}$

$y'=2x^{2-1}$

3. The slope at any point $\displaystyle x$ on the curve $\displaystyle y = x^2$ is given by $\displaystyle \frac{dy}{dx} = 2x$.

So at the point $\displaystyle x = 7$, the slope is $\displaystyle 2\cdot 7 = 14$.

Equation of tangent: $\displaystyle y = mx + c$.

You have a coordinate that lies on your tangent $\displaystyle (x, y) = (7, 49)$ and the slope $\displaystyle m = 14$. Substitute and solve for $\displaystyle c$, then you can write the equation of your tangent line.

4. How well do you understand the concept behind differentiation?

If $y=x^2,$

$\displaystyle\frac{dy}{dx} = 2x$

$
\displaystyle\frac{dy}{dx}$
is 'the rate of change of y with respect to x'. It is used to tell us the equation of the gradient of the curve at any point.

In the question, $\frac{dy}{dx} = 2x$. So take any point on your curve. Let's say $(3,9)$ This has x co-ordinate 3.

Substituting this into $\frac{dy}{dx} = 2x$ gives $\frac{dy}{dx} = 6$. What this means is that at the point $(3,9)$ the gradient of the curve $y=x^2$ is 6.

Also, I don't think that $2x$ is the answer to that question.

We have a point (7,49) and we know that $\frac{dy}{dx} = 2x$. So the gradient of the tangent is $2\times 7 = 14$.

So we know that when x = 7, y = 14 and m is 14. Substitute this into y = mx + c to find c and you'll find the equation of the tangent.

5. from first principles ...

$\displaystyle f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$

$f(x) = x^2$

$f(x+h) = (x+h)^2$

$\displaystyle f'(x) = \lim_{h \to 0} \frac{(x+h)^2 - x^2}{h}$

$\displaystyle f'(x) = \lim_{h \to 0} \frac{x^2+2xh+h^2 - x^2}{h}$

$\displaystyle f'(x) = \lim_{h \to 0} \frac{2xh+h^2}{h}
$

$\displaystyle f'(x) = \lim_{h \to 0} \frac{h(2x+h)}{h}
$

$\displaystyle f'(x) = \lim_{h \to 0} 2x+h = 2x$