Confusing the gradient value with the y-value

To write the equation of a tangent, you need two ingredients at the given x: the point on the curve and the gradient there. The point comes from the original curve, substitute $x$ into $y$.

The gradient comes from the derivative, substitute the same $x$ into $\frac{dy}{dx}$. The confusion is treating these two numbers as interchangeable.

Students substitute into $\frac{dy}{dx}$, get a value, and then reuse that same value as the y-coordinate of the point, so the gradient becomes the height. Others do the reverse, feeding the curve's y-value into the tangent as though it were the slope.

It is tempting because both come from putting the same $x$ into an expression, and the two expressions look similar. But one measures how high the curve is and the other measures how steep it is; they answer completely different questions.

Find the tangent to $y=x^{2}-3x+4$ at $x=2$. Differentiate to get the gradient function $\frac{dy}{dx}=2x-3$, and at $x=2$ this is $2(2)-3=1$, so the gradient is $1$.

The number $1$ is the slope, not the height. The y-coordinate must come from the original curve, not from the derivative.

Substitute $x=2$ into $y=x^{2}-3x+4$: $y=4-6+4=2$.

Now put both into $y-y_{1}=m(x-x_{1})$: $y-2=1(x-2)$, which simplifies to $y=x$. Checking, $x=2$ gives $y=2$ on both the curve and the tangent, and the slope is $1$, so the line touches the curve at the right point with the right steepness.

1. 1

   Get the y-coordinate from the curve

   Substitute the given $x$ into the original equation $y=\ldots$. This number is the height of the point, label it $y_{1}$.

2. 2

   Differentiate the curve

   Find $\frac{dy}{dx}$ as a function of $x$. This is the gradient function, not a coordinate.

3. 3

   Get the gradient from the derivative

   Substitute the same $x$ into $\frac{dy}{dx}$. This number is the slope, label it $m$.

4. 4

   Keep the two numbers labelled

   Write the point as $(x_{1},y_{1})$ with $y_{1}$ from the curve, and $m$ separately from the derivative. Never let one stand in for the other.

5. 5

   Assemble the line

   Substitute into $y-y_{1}=m(x-x_{1})$ and simplify. Check the point satisfies your final equation.

Our teachers make the two substitutions visibly separate, one box for 'height from the curve', one for 'slope from the derivative', so the numbers never blur into each other. In a one-to-one lesson we catch the exact moment a gradient gets reused as a coordinate and re-anchor it by asking which question each number answers.

Because the paper awards method marks, labelling the point and the slope clearly protects your marks even if a later step slips. Teachers at spmaddmath.com.my are experienced, and lessons are online and taught in English.

Message us on WhatsApp to arrange a one-hour paid class from RM50/hr at the teacher's rate.