# Show that the gradient of the secant line to the curve #y=x^2+3x# at the points on the curve where #x=3# and #x=3+h# is #h+9#?

We have:

Hence, the gradient of the secant line can be found using:

In summary

It should be evident that the definition of the derivative leads directly to this final result. We can confirm this by applying our calculus knowledge, as follows:

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To find the gradient of the secant line, we need to calculate the slope between two points on the curve.

Let's start by finding the coordinates of the two points on the curve. When x = 3, the corresponding y-value can be found by substituting x = 3 into the equation y = x^2 + 3x:

y = (3)^2 + 3(3) = 9 + 9 = 18

Now, let's find the coordinates of the second point when x = 3 + h:

y = (3 + h)^2 + 3(3 + h) = 9 + 6h + h^2 + 9 + 3h = h^2 + 9h + 18

Now that we have the coordinates of the two points, we can calculate the slope (gradient) of the secant line using the formula:

Gradient = (change in y) / (change in x)

(change in y) = (h^2 + 9h + 18) - 18 = h^2 + 9h (change in x) = (3 + h) - 3 = h

Therefore, the gradient of the secant line is:

Gradient = (h^2 + 9h) / h = h + 9

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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