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# Derivatives

### Derivatives

Calculate y' (y prime) 1. y = cos(tanx) 2. y = e^(-1)*(t^2-2t+2) 3. y = sin^(-1)*(e^x) 4. y = x^r*e^(sx) 5. y = 1/(sin(x-sinx)) 6. y = ln(csc5x) 7. x^2 cosy + sin2y = xy 8. y = ln(x^2*e^2) 9. y = sec(1+x^2) 10. y = (cosx)^x

### Derivatives using product rule, quotient rule, or chain rule.

Find the derivative of each expression, using the product rule, quotient rule, or chain rule. 1. P= e^(2x)/x 2. B= square root of sin * square root of x 3. Find dy/dx using implicit differentiation. (3xy + 1)^5 = x^2

### Polar Coordinates : Solving Derivatives and Circuit Problem

Please see the attached file for full problem description. (a) By making the substitution y = z/x^4, or otherwise, reduce the equation dy/dx +4y/x =sinx/x2 to an equation in which the variables are separable. Solve the equation if y = 0 when x = pi/2 (b) In a circuit di/dt=K(E-Ri) and i=0 when t=0. Find i in

### First Principle in evaluating derivatives

(A) Find and simplify the difference quotient for G(X)=1/x^2. HINT: After finding the difference quotient, simplify by using an LCD to combine the fractions. (B) Using the answer above, find the value of the difference quotient at x=1 with an h=.1 C) Sketch a graph of G(x). Mark the point(1,G(1)) on the graph. Sketch a

### Going from First to Second Derivative

I have a first derivative and a second derivative, how do you get from the first to the second, I can't solve it. Please see the attached file for the fully formatted problems. M'(t) = pe^t/(1 - t^tq)^2 M''(t) = pe^t(1 + qe^t)/(1 - (e^t)q)^3

### Orthogonal trajectories, family of curves 2cy+x^2=c^2, c>0

Please see attachment. Please help me solve problem in its entirety. I'm having trouble, most of all, solving the D.E. Thanks. :)

### 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

### Differentiation sample problem

See attached file for full problem description. y = 1/x^4 - 3x^2 + 2x - 1.6

See attachment

### Log differentiation

Find the derivative dy/dx if y = x^(x+3)

### Partial derivatives

The heat transfer in a semi-infinite rod can be described by the following PARTIAL differential equation: &#8706;u/&#8706;t = (c^2)&#8706;^2u/&#8706;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 using the product rule and the chain rule with the exponential function

Find the derivative of the function y=x^2e^(3x) See Attachment for a cleaner version of the question.

### Real Analysis Functions with Equal Derivatives

Let f be a function given by x + 2 if x < 0 f(x) = x if x >= 0 Is there a function g: R ---> R such that g'=f? *be careful applying definition of the derivative

### Differentiation : Existence of Solutions

I want to prove, for the numbers a and b, that the following equation has exactly three solutions if and only if 4a^3 + 27b^2 < 0: x^3 + ax + b = 0, x in R

### Derivatives and differentiation: Rate of Change - Surface Area of a Cube

At what rate is the surface area of a cube changing the edge measures 5 inches and is changing at a rate of 2 in/min. GIVEN (A=6*s^2)

### 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.

### Find both first partial derivatives.

A) z=y^3-4xy^2-1 B) first partial derivative WRT x,y,z for : w=3xz/(x+y)

### 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.

### Derivative of a Function: Product Rule

Use the product rule find derivative of h(t)= cubed root of t * (t^2+4)

### Partial Derivatives

Find the first partial derivative with respect to x, y, z. w = 3xz / x+y

### 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)

### Calculate the derivatives of wave equations to determine the direction of travel.

The equation for a wave moving along a straight wire is: (1) y= 0.5 sin (6 x - 4t) To look at the motion of the crest, let y = ym= 0.5 m, thus obtaining an equation with only two variables, namely x and t. a. For y= 0.5, solve for x to get (2) x(t) then take a (partial) derivative of x(t) to get the rate of change of

### Equating the function of a reversal tangent.

What is the equation of the reversal tangent of the following function: f(x)=-(x^3)+9x^2-(29x)+35

### Implicit differentiation : the chain rule and product rule.

Use implicit differentiation to find dy/dx if y^2 + 3xy + x^2 + 10 = 0 (1) where y is a function of the independent variable x.

### Derivatives from first principles using standard theorems on trigonometric limits.

Using the definition of the derivative and any standard limiting theorems, show that the derivative of (sinx)^2 is sin(2x).