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Derivatives

Rate of change

Suppose that the temperature at the point (x, y, z) in space (in degrees Celsius) is given by the formula: W= 100 - x^2 - y^2 - z^2. The units in space are meters. (a) Find the rate of change of temperature at the point P(3, -4, 5) in the direction of the vector v=3i - 4j + 12k. (b) In what direction does W increase most rapi

Directional derivative

Find the directional derivative of f at P in the direction of v; that is find D_u f(P), where u=v/{v}: f(x, y, z)= ln(1 + x^2 +y^2 - z^2) ; P(1, -1, 1), v=2i - 2j -3k

Chain rule

Write chain rule formulas giving the partial derivative of the dependent variable p with respect to each independent variable: p=f(x, y, z); x=x(u, v), y=y(u, v), z=z(u, v)

Partial derivative question

I am taking a course by distance, and my professor provided an example of how to create a Hessian matrix using partial derivatives. He gave another example that just had the solution for us to try on our own. I think that I am somehow not taking the second order partial derivative right. The attached file has the professor

Partial derivatives

The heat transfer in a semi-infinite rod can be described by the following PARTIAL differential equation: ∂u/∂t = (c^2)∂^2u/∂x^2 where t is the time, x distance from the beginning of the rod and c is the material constant. Function u(t,x) represents the temperature at the given time t and p

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)