Use the function f(x) = 4x + 5x^2 to find the following. (a)...

Use the function $f(x) = 4x + 5x^2$ to find the following. (a) Use the limit definition of the derivative to find $f'(x)$. $f'(x) =$ (b) Find the equation of the tangent line at $x = 2$.

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00:01 So in order to do the limit definition of the derivative, you also have to acknowledge what f of x plus h is.

00:09 And that would be two over.

00:11 You replace the x with x plus h.

00:14 So what they're saying is the derivative of x, let's write down the formula, is f of x plus h minus f of x all over h.

00:24 I forgot to write the limit.

00:26 The limit as h approaches zero.

00:30 So let's do that.

00:32 As h approaches 0, f of x plus h was this piece over here.

00:39 So that's 2 over x plus h.

00:42 And then f of x was this piece here.

00:45 So that's 2 over x, all over h.

00:48 And it's nice to rewrite f prime of x, but you probably don't have to.

00:53 So what i would do next is work with the exponent and get the same denominators.

00:58 So think back to probably a long time.

01:01 Ago when you got the same denominators, you had to multiply them together.

01:06 So what you had to do is this piece right here was missing in x.

01:10 So you had to multiply the top by x as well.

01:13 So that's 2x.

01:15 And then this piece over here is missing the x plus h.

01:18 So you have to multiply the top by x plus h.

01:21 And not only do i distribute the negative, the 2, but i also distribute that negative...

Question

Use the function $f(x) = 4x + 5x^2$ to find the following. (a) Use the limit definition of the derivative to find $f'(x)$. $f'(x) =$ (b) Find the equation of the tangent line at $x = 2$.

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