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I am pretty thick and need questions to be specific or I do not know what they want. Do they want me to give a random example for x? eg the slope at x=7 is 5x?

Ray Kay
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    No. They want you to define $f'(x)$. – Daniel W. Farlow Mar 13 '15 at 09:26
  • No, a formula for the slope in terms of x. If $y = x^2$ then the slope of the tangent to this curve at x = the derivative of y with respect to x = the rate of change of y with respect to x = 2x. – Paul Mar 13 '15 at 09:27

2 Answers2

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Given a function $f( x)$. Then the graph for this function is:

$\phi(x) =(x,f(x))$.

Evaluate derivative:

$D\phi(x) =(1,f'(x))$

and set:

$ t_{x_0}(\lambda)=\phi(x_0)+\lambda D\phi(x_0)$

Now we have got a parametrization for the tangent at the point $\phi(x_0)$ on the graph

We can rewrite this as follows. Set

$x(\lambda)=x_0+\lambda $

$y(\lambda)=f(x_0)+\lambda f'(x_0)$

Eliminating for $\lambda$:

$$ T_{x_0}(x)=f(x_0)+f'(x_0)(x-x_0)$$

we have got the tangent-line for any point $(x_0,f(x_0))$.

Frieder
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They want an expression in terms of $x$ that can be used to find the gradient at any point.

So for example, if the curve is $y=3x^2$, then the gradient function is $6x$. That is what they are asking for.

Although they aren't asking for it, you can use this to find the gradient at a specific point. So at $(3,27)$ the gradient is $18$.

tomi
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