Question

Using the definition of the derivative, find $f'(x)$. Then find $f'(-1)$, $f'(0)$, and $f'(3)$ when the derivative exists. $f(x) = 5x^3 + 8$ To find the derivative, complete the limit as h approaches 0 for $ \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$ $\lim_{h \to 0} \frac{15x^2 + 15xh + 5h^2}{h}$ Find $f'(x)$ using the definition of the derivative. $f'(x) = 15x^2$ Find $f'(-1)$ if the derivative exists. Select the correct choice below and, if necessary, fill in the answer box to complete your A. $f'(-1) = 15$ (Simplify your answer.) B. The derivative does not exist. Find $f'(0)$ if the derivative exists. Select the correct choice below and, if necessary, fill in the answer box to complete your c A. $f'(0) = $ (Simplify your answer.) B. The derivative does not exist.

          Using the definition of the derivative, find $f'(x)$. Then find $f'(-1)$, $f'(0)$, and $f'(3)$ when the derivative exists.
$f(x) = 5x^3 + 8$
To find the derivative, complete the limit as h approaches 0 for $
\lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$
$\lim_{h \to 0} \frac{15x^2 + 15xh + 5h^2}{h}$
Find $f'(x)$ using the definition of the derivative.
$f'(x) = 15x^2$
Find $f'(-1)$ if the derivative exists. Select the correct choice below and, if necessary, fill in the answer box to complete your
A. $f'(-1) = 15$ (Simplify your answer.)
B. The derivative does not exist.
Find $f'(0)$ if the derivative exists. Select the correct choice below and, if necessary, fill in the answer box to complete your c
A. $f'(0) = $ (Simplify your answer.)
B. The derivative does not exist.

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Using the definition of the derivative, find f'(x). Then find f'(-1), f'(0), and f'(3) when the derivative exists.
f(x) = 5x^3 + 8
To find the derivative, complete the limit as h approaches 0 for limh → 0(f(x+h) - f(x))/(h)
limh → 0(15x^2 + 15xh + 5h^2)/(h)
Find f'(x) using the definition of the derivative.
f'(x) = 15x^2
Find f'(-1) if the derivative exists. Select the correct choice below and, if necessary, fill in the answer box to complete your
A. f'(-1) = 15 (Simplify your answer.)
B. The derivative does not exist.
Find f'(0) if the derivative exists. Select the correct choice below and, if necessary, fill in the answer box to complete your c
A. f'(0) = (Simplify your answer.)
B. The derivative does not exist.

Added by Michael H.

Calculus: Early Transcendentals

James Stewart 8th Edition

Instant Answer

Solved by Expert William Semus

12/28/2021

Step 1

Given: f(x) = 5x^3 + 8 To find the derivative, we need to find the limit as h approaches 0 for the expression: lim(h->0) [5(x+h)^3 + 8 - (5x^3 + 8)] / h Expanding and simplifying the expression: lim(h->0) [5(x^3 + 3x^2h + 3xh^2 + h^3) + 8 - 5x^3 - 8] / Show more…

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Using the definition of the derivative, find f^(')(x). Then find f^(')(-1),f^(')(0), and f^(')(3) when the derivative exists. f(x)=5x^(3)+8 To find the derivative, complete the limit as h approaches 0 for . lim_(h->0)15x^(2)+15xh+5h^(2) Find f^(')(x) using the definition of the derivative. f^(')(x)= Find f^(')(-1) if the derivative exists. Select the correct choice below and, if necessary, fill in the answer box to complete you A. f^(')(-1)=15 (Simplify your answer.) B. The derivative does not exist. Find f^(')(0) if the derivative exists. Select the correct choice below and, if necessary, fill in the answer box to complete your 0 A. f^(')(0)= (Simplify your answer.) B. The derivative does not exist. K Using the definition of the derivative,find fx.Then find f-1,f'0),and f'3when the derivative exists fx=5+8 h lim152+15xh+5h2 h-0 Find f'x using the definition of the derivative. fx=152 Find f-1if the derivative exists. Select the correct choice below and,if necessary,fll in the answer box to complete yol A.f-1=15Simplify your answer. B.The derivative does not exist Find f0 if the derivative exists. Select the correct choice below and, if necessary,fill in the answer box to complete your OA.f0=Simplify your answer. OB.The derivative does not exist.

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Transcript

00:01 So in this question, we're asked to use the definition of the derivative to find f prime of x if f of x is equal to negative x squared plus 3x minus 5.

00:11 So first, we remember what the limit definition of the derivative is.

00:17 F prime of x is equal to the limit.

00:21 As h approaches 0 of f of x plus h minus f of x all being divided by h.

00:34 What is f of x plus h this time? f of x plus h means i go to my f of x and i replace every instance of x with the quantity of x plus h.

00:51 So that i have negative the quantity of x plus h being squared plus 3 times the quantity of x plus h minus 5.

01:06 That's f of x plus h.

01:09 Now from this, i subtract f of x.

01:13 I subtract the entire quantity of f of x, so that's got to go in parentheses.

01:21 And then this is all being divided by h.

01:27 Now, we have a bunch of algebra to do.

01:30 We're going to have to multiply things out and combine like terms.

01:36 So let's see, if i square out the x plus h quantity squared, i get xx.

01:43 Squared plus 2xh plus h squared.

01:49 Distribute the 3 plus 3x plus 3h minus 5.

01:58 Distribute the negative side plus x squared minus 3x plus 5.

02:09 All of this is being divided by each.

02:13 Now notice a few things start to cancel for me.

02:15 The negative 5 and the positive 5 cancel.

02:18 The positive 3x and the negative 3x they cancel as well.

02:25 In my next line, i'm going to maintain my limit as h approach is 0.

02:31 And i distribute the negative sign at the beginning.

02:36 Negative x squared minus 2xh, minus h squared, plus 3h, plus x squared, all being divided by h...

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