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Find equation of tangent to a curve

The tangent to a curve is a line which touches the curve at a point without intersecting it at that point so the gradient of the curve at that point and the gradient of the tangent are the same. So we can work out the point the tangent passes though and the gradient of the tangent from the equation of the curve, which will give us enough information to find the equation of the tangent.

Example y=x2
Find the equation of the tangent to the curve $y=x^2$ when x=4?

To do this we first need to find the gradient of the curve which we can do by differentiating it.
$\frac{d}{dx}(x^2) = 2x$
so at the point x=t the gradient is 2t.

From this we can get a general equation for the tangent using the equation for the gradient of a straight line
grad = \$latex \frac{y – y_1}{x – x_1}
to get the general equation for the tangent at the point x=t by substituting x1=t, y1=t^sup>2 and m=2t
$2t = \frac{y - t^2}{x - t} \rightarrow 2xt - 2t^2 + t^2 = y \rightarrow y=2xt-t^2$

Then we can substitute in t=4 to find the equation of the tangent when x=4 to get
$y=8x-16$