What is Power rule: Definition and 42 Discussions

In calculus, the power rule is used to differentiate functions of the form




{\displaystyle f(x)=x^{r}}
, whenever


{\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.

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  1. C

    Power Rule Proof: Get Help with Line 3 to Line 4

    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!
  2. karush

    MHB S8.2.6.3 differenciate by separation and power rule

    $\tiny{s8.2.6.3}$ Find y' $\sqrt{x}+\sqrt{y}=1$ \begin{array}{lll} \textit{isolate }y &\sqrt{y}=1-\sqrt{x} &(1)\\ \\ \textit{square both sides} &y=(1-\sqrt{x})^2 &(2)\\ \\ \textit{differentiate both sides} &y'=2\left(1-\sqrt{x}\right)\left(-\dfrac{1}{2\sqrt{x}}\right)&...
  3. karush

    MHB 215 AP Calculus Exam Problem Power Rule

    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...
  4. YoungPhysicist

    Understanding the Derivative of a Polynomial with Exponent 10

    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...
  5. YoungPhysicist

    I Integral power rule explanation

    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...
  6. C

    Integration: Can't understand when to use Power rule.

    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...
  7. BiGyElLoWhAt

    Excellent video series raises good question:

    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...
  8. J

    Power rule of an integration [beginner]

    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!
  9. Greg Bernhardt

    What is the power rule for taking derivatives of polynomial functions?

    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...
  10. A

    Proof of integration power rule

    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}...
  11. D

    Can I Use the Power Rule to Get the Derivative Here?

    can i use the power rule to get the derivative here? f ' (x) = 3x^2 - 2(2x^1) + 1
  12. R

    General Power Rule: Proof & Real Values of x?

    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...
  13. Mandelbroth

    Integration by Parts versus the Power Rule

    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 \...
  14. Astrum

    Proof of the Power Rule - Stuck at the End

    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}...
  15. T

    Proof by Induction of the Power Rule of Differentiation

    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...
  16. T

    Prove the case r= -n (n is a positive integer)of the general power rule

    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).
  17. L

    What is the uncertainty in a squared quantity?

    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...
  18. C

    The power rule and the chains rule

    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
  19. B

    Partial derivatives and power rule

    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
  20. P

    Derivative of Fraction using power rule

    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?
  21. N

    Why does the power rule work? [answer is probably :/ obvious]

    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...
  22. 5

    Prove Power Rule by Math Induction and Product Rule

    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) =...
  23. B

    Mathematical Induction: Power Rule for Differentiation

    Homework Statement Prove that \frac{d}{dz}z^n = nz^{n-1}\;\;\; \forall n\in\mathbb{N} using the Product Rule for differentiation and mathematical induction. Homework Equations \frac{d}{dz} f(z) = \lim_{\Delta z\to 0} \frac{f(z+\Delta z) - f(z)}{\Delta z} \frac{d}{dz}[f(z)g(z)] = f\,'(z)g(z) +...
  24. Femme_physics

    Skipping the proof of the power rule how inadvisable?

    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...
  25. D

    Interpretation of power rule for integration applied to 1/x

    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
  26. M

    A problem with a power rule proof

    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?
  27. C

    Trigonometric derivative: power rule + product rule

    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...
  28. X

    How can I use the power and quotient rule to solve this problem?

    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)] -...
  29. D

    Proof of Power Rule (Definite Integral) for Negative Integers

    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 =...
  30. D

    How can the power rule be used to differentiate fractions in a simpler way?

    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...
  31. L

    Power rule valid for any complex exponents?

    Is the power rule in calculus true for any complex exponents? i.e. does d/dz z^c = cz^(c-1) true for any c, even when c is something like i or 3i-2?
  32. H

    Proving the power rule by induction

    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...
  33. B

    Proof of power rule of limit laws

    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...
  34. A

    Proof of Power Rule for 2 variables

    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
  35. B

    Can the Power Rule be applied to all rational numbers in logarithms?

    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)...
  36. F

    Power Rule Question: Why Did 1-2x Become 2x-1?

    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)??
  37. S

    Solving a Derivative Problem Using the Power Rule

    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...
  38. P

    Power rule for antiderivatives

    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...
  39. M

    Prove the Power Rule (calculus)

    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...
  40. M

    Prove Power Rule: Implicit Differentiation w/ Rational Exponents

    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...
  41. F

    Power rule and product rule problem

    (2x+3)^3(x-4) How do I differentiate this?
  42. P

    Finding Derivatives: Power Rule vs Chain Rule

    In finding a derivative of a value, how do you know whether when to use the power rule or the chain rule? can anyone please tell me?