In calculus, the power rule is used to differentiate functions of the form
f
(
x
)
=
x
r
{\displaystyle f(x)=x^{r}}
, whenever
r
{\displaystyle r}
is a real number. Since differentiation is a linear operation on the space of differentiable functions, polynomials can also be differentiated using this rule. The power rule underlies the Taylor series as it relates a power series with a function's derivatives.
For this proof,
I am unsure how they got from line 3 to line 4.
If I simplify and collect like terms for line 3 I get ##f'(a) = 4a^{n-1}##
Would some please be able to help?
Many thanks!
If $y=(x^3-cos x)^5$, then $y'=$(A) $\quad 5(x^3-\cos x)^4$(B) $\quad 5(3x^2+\sin x)^4$(C) $\quad 5(3x^2+\sin x)^4$(D) $\quad 5(x^3+\sin x)^4(6x+\cos x)$(E) $\quad 5(x^3+\cos x)^4(3x+\sin x)$
ok I am sure this could be worded better. but I think many students take these tests and are not used...
Homework Statement
##f(x) = (5x+6)^{10} , f'(x)=?##
Homework Equations
##\frac{d}{dx}x^n = nx^{n-1}##?
3. The Attempt at a Solution [/B]
I do know the solution ##f'(x) = 50(5x+6)^9##,but I don't know how this solution came to be.I downloaded this problem from the web and it only comes with...
I am new to the world of calculus and the first thing that I learned is how to calculate the area under the range of a polynomial function, like:
$$\int_1^3 x^2 \,dx$$
when I take the intergal of ##x^2##, I get ##\frac{x^3}{3}##due to the power rule,
but it doesn’t make sense to me,why would...
Homework Statement
Hi I have a really hard time understanding when and how to use the Power rule when integratingMy book states that if u is a function of x, then the power rule is given by:
∫urdu = (ur+1)/(r+1)+c
First: I do understand that if
u=2x
Then the differential
du=u'(x)dx =2
So...
www.youtube.com/watch?v=oW4jM0smS_E
That's the video I'm referencing in particular, but 1 and 3 are necessary prereqs if you're new to the matter (as I am).
He goes through and derives the product rule and power rule for polynomials using algebra.
My question is this: why don't we teach...
So I stumbled upon ∫1/(x^4) , and by applying the power rule , the answer is: -1/(3x^3)
Why's that? Sorry for bothering you guys with such a beginner question!
Definition/Summary
A method used to take the derivative of a polynomial function.
Equations
\frac{d}{dx} x^{n} = nx^{n-1}
Extended explanation
Power rule applies to a function of the form x^{n}, where x is the variable and n is a constant. Used in combination with the sum and...
Hey, I was just wondering if there was a way to prove the power rule for integration using the definition of a definite integral. And I don't mean using the proof for the differentiation power rule, I mean is it possible to derive \displaystyle\large\int_a^b x^c=\frac{b^{c+1}-a^{c+1}} {c+1}...
Hi all, regarding the proof of the general power rule,
If we let y = x^r, then \ln y = r\ln x, and then by implicit differentiation
\frac{y'}{y} = \frac{r}{x},
and thus it follows that
y' = \frac{ry}{x} = \frac{rx^r}{r} = rx^{r-1}.
But the statement \ln y = r\ln x also requires x>0, so...
Recently, a friend of mine asked for help on their calculus homework. The problem was to find \int cos(ln \ x) \ dx. However, I've never gotten around to memorizing the derivatives and integrals of the trig functions.
I know that you can do it using integration by parts, with \int cos(ln \...
Homework Statement
Prove that the following
f'(x)=nx^{n-1} if f(x)=x^{n}Homework Equations
Binomial theorem, definition of the derivative The Attempt at a Solution
f'(x)=lim_{h\rightarrow0}\frac{f((x+h)^{n})-f(x)}{h}
We need to expand the (x+h)^2 term now
\sum^{n}_{k=0}{n\choose k}...
Homework Statement
Okay, the concept here is to use induction to prove that for n, (f1 x f2 x ... x fn-1 x fn)' = (f'1 x f2 x ... x fn) + (f1 x f'2 x ... x fn) + ... + (f1 x f2 x ... x f'n).
2. Homework Equations / 3. The Attempt at a Solution
I solved the initial step, which was quite...
Prove that:
d/dx x^-n = -nx^-n-1
Use the factorization of a difference of nth powers given in this section (not using quotient rule)
My attempt gets me from the definition of the derivative to (1/n^(n-1)) n times... I need the negative. I get nx^(-n-1) instead of -nx^(n-1).
Homework Statement
D= A +/-ΔA
D= 5.160 +/- 0.01 cm
D^2= 26.6 +/- 0.1 cm^2
Homework Equations
for the power rule uncertainty
:
A ((ΔA/A) + (ΔA/A) )
So then its (5.160)( (0.01/5.16)(2)) = 0.004
The Attempt at a Solution
im getting 0.004 as the absolute uncertainty but the...
Homework Statement
How do I know for sure when to use the power rule instead of the chain rule and vice versa?
Homework Equations
The Attempt at a Solution
Homework Statement
∂f/∂x (xy -1)2 = 2y(xy-1)
The Attempt at a Solution
I would think the answer would be
2(xy-1)
I don't understand where the y comes from in 2y
The problem:
f(x)=3-3/5x
So I'm perfectly fine with finding the derivatives with stuff but I wasn't sure about this one. Would this be 0 because there is a three in the numerator and no x?
Or would it be 3-1/51-1=3-1=2?
Can someone explain to me in intuitive terms why the trick of bringing the the exponent out front and then reducing the power by 1 works?
Solving problems using the Limit definition of a derivative (where we take a secant line closer and closer to a point) makes great intuitive sense, but I...
Homework Statement
Use the Principle of Mathematical Induction and the Product Rule to prove the Power Rule when n is a positive integer.
Homework Equations
Dxxn = nxn-1
Dx(fg) = fDxg + Dxfg
The Attempt at a Solution
In summary,
Dxxn = nxn-1
Dxxk = kxk-1
Dxxk+1 = (k+1)x(k+1)-1
Dx(xkx) =...
It just seems too complicate to bother to comprehend. I know you guys are math/physics fundies, but I'm really just trying to get the hang of the material and be able to solve problems, I'm not looking for some sort of cosmic math super-understanding. Should I really bother with the proof of the...
We all know
\int \frac{1}{x} dx = ln(x) + c
but if you try to apply the power rule for integration:
\int x^n dx = \frac{x^{n+1}}{n+1} + c
you get
\int x^{-1} dx = \frac{x^0}{0}
What can you learn from this/what does this mean?
David
Homework Statement
y=xn
<=>
ln(y)=nln(x)
<=>d/dx ln(y) = d/dx n ln(x) <=> y'/y=n/x
<=> y'=y*n/x=x^n*x/x=nx^(n-1)
Homework Equations
The Attempt at a Solution
but doesn't this proof only hold for positive x? because ln a negative number is undefined or am I missing something?
Homework Statement
Derive the following:
y = (cosxsin2x)-2
2. The attempt at a solution
Basically I saw this as a power rule with two products in the middle.
So y = -2 (cos2cos2x-sinxsin2x)-1
But the correct answer is completely different, it's:
4(3sin2x - 1) all over...
Hi everyone,
I have been trying to do this problem in both ways but I can't get the same answer the book says. This is the problem:
x/ sqrt (x^2 +1)
With quotient rule I got until the point I have [(x^2 +1)^1/2 - x^2/(x^2 +1)^1/2]/(x^2 +1)
And with power rule I have [1/sqrt(x^2 +1)] -...
This is not actually a homework problem. Rather, it is a problem from Courant and Robbins' What is Mathematics?, Chapter 8: "The Calculus", page 409-410.
Homework Statement
Prove that for any rational k =/= -1 the same limit formula, N → k+1, and therefore the result:
∫a to b xk dx =...
Its more a simplifying problem...
I was trying to differentiate this using definition principal
1-x^1/2
But I got stuck here :
(1-(x+h)^1/2 - (1-x^1/2))/h
I mean how do you explan something to the power of half or infact any fraction? I know I can change it to 1/sqrt(x+h)... but...
Homework Statement
If a is a natural number, prove by induction that
y = [g(x)]^a => y' = a[g(x)]^(a-1) * g'(x)
Homework Equations
Let a = 2
y' = (2)[g(x)]^(2-1) g(x)
= 2g(x)g'(x)
Let a = 3
y' = (3)[g(x)]^(3-1) g(x)
= 3g(x)^2 * g'(x)
Let k be any natural...
Homework Statement
Power Rule: If r and s are integers with no common factor and s=/=0, then
lim(f(x))r/s = Lr/s
x\rightarrowc
provided that Lr/s is a real number. (If s is even, we assume that L>0)
How can I prove it?
Homework Equations
The Attempt at a Solution
I heard that...
Homework Statement
u^n (x,y)=nu^(n-1) (x,y) u' (x,y)
Homework Equations
The Attempt at a Solution
can i set f (x,y)=u^n (x,y)
lnf =lnu^n
lnf=nlnU
f'/f=n/U
f'=fn/U -(U(^n) )(n/U)
after that i don't know how to continue and is there a better way to prove it
All information, including the problem, is attached. So far I think I've proven by induction that log (a^r) = r log (a) whenever r is an integer, but I need to prove this for all rational numbers r = p/q .
We're working with the functional equation that has the property that f(xy) = f(x)...
I just did a problem that made use of the power rule. Here is the final step:
(1-2x)
-------------
-(x-x^2)^5/4
=
(2x-1)
-------------
-(x-x^2)^5/4
Why did 1-2x become 2x-1 (2x-1 is the answer according to my book)??
I tried this derivative problem, but the back of the book shows a different answer then what I got. Can someone explain what I'm doing wrong.
3T^5 -5T^.5 + \frac{7}{T}
So I did this:
15T^4 - 2.5T^-.5 - 7T^-8
It's the last part I got wrong I'm not sure why. I converted 7/T to T^-7. Is that...
I am taking an Architectural Geometry class, and have only had Precal. We just started antiderivatives (I understand regular derivatives), and had a question:
I have to find the antiderivative of
(-5/12 x^4) + (10/3 x^3) - (103/12 x^2) + (23/3 x)
I think I use the power rule for...
Ok guys, I'm new here and I need some help with a math problem...
The problem asks me to prove the power rule ---> d/dx[x^n] = nx^(n-1) for the case in which n is a ratioinal number...
the one stipulation is that I have to prove it using this method: write y=x^(p/q) in the form y^q = x^p...
Ok guys, I'm new here and I need some help with a math problem...
The problem asks me to prove the power rule ---> d/dx[x^n] = nx^(n-1) for the case in which n is a ratioinal number...
the one stipulation is that I have to prove it using this method: write y=x^(p/q) in the form y^q = x^p...