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Derivatives

Differentiation

Prove that if f(x) = x^alpha, where alpha = 1/n for some n in N (the natural numbers), then y = f(x) is differentiable and f'(x) = alpha x^(alpha - 1). Progress I have made so far: I have managed to prove, (x^n)' = n x^(n - 1) for n in N and x in R both from the definition of differentiation involving the limit and the binomial

Partial derivatives

Find partial deriv's w/r and w/theta using appropriate chain rule for : w=the square root of (25-5x^2-5y^2), where x=r cos theta, y= r sin theta.

Maclaurin Series

Given The Maclaurin series for the inverse hyperbolic tangent is of the form x+x^3/3+x^5/5...x^7/7. Show that this is true through the third derivative term.

True or False Derivative Questions

True or False and Why? 1.) If f(x) = g(x) + c, then f'(x) = g'(x) 2.) If y = x/pi, then dx/dy = 1/pi 3.) If f(x) = 1/x^n, then f'(x) = 1/(nx^n-1)

Basic Derivatives

1.) Find the derivative of the function: a.) f(x) = x + 1/x^2 b.) f(x) = (2/3rd root of x) + 3 cos x 2.) Find equation of tangent line to the graph of f at the indicated point: a.) y = (x^2 + 2x)(x + 1) ; (1,6)

Derivatives: Chain Rule

Please see the attached file for the fully formatted problems. For a composite function f(x) = g(u(x)) state the chain rule for the derivative f(x). For each of the following functions, compute the derivative, simplifying your answers. f(x)=ln(1 + x^2) f(x)= sin(x^2) f(x) = (sin x)^2 (a) For a composite function f(x)

Definition of derivative

Let f(x) be a continuous function of one variable. a) Give the definition of the derivative. b) Use this definition to find the derivative of f(x)=x^2+2x-5 c) Evaluate f'(2)

Proof Regarding a Twice Differentiable Function

Context: We are learning Rolle, Lagrange, Fermat, Taylor Theorems in our Real Analysis class. We just finished continuity and are now studying differentiation. Question: Let f: [a,b] --> R, a < b, twice differentiable with the second derivative continuous such that f(a)=f(b)=0. Denote M = sup |f "(x)| where x is in [a,b]

Finding a rule for a sequence.

Find the rule for the sequence below. square 2x2 =10 3x3=40 +30=20 difference 4x4=90+50=20 difference 5x5=160+70=20 difference

Finding a Polynomial

Find a polynomial p so that: p''(t)+3p'(t) + 2p(t) = (t^2)-2 for all numbers t. (note: p''= p double prime and t^2 = t raised to the power of 2)

Using related rates to answer a pulley question.

Two carts A and B are connected by a rope 39 feet long that passes over pulley P. The point Q is on the floor directly beneath P and between the carts. Cart A is being pulled away from Q at a speed of 2 ft/sec. How fast is cart B moving toward Q at the instant cart A is 5 feet from Q? Express solution using related rate n

Limits and derivatives

Find the derivative. a) f= 4-sqrt(x+3) b) f= (x+1)/(2-x) See attachment below for additional information.