Suppose that f is a differentiable function such that f(4)=4 and f'(4)=-2. Let g(x)=f(xsquare root of x). Evaluate g'(4).

Q: If ƒ(x) = x^n, n>=1, show from the definition of the derivative that ƒ′(0) = 0.

A: Given,

Q: Use the limit definition of the derivative to prove the following special case of the Product Rule:

A: The derivative of the function is the rate of change of the value of the function when the value of…

Q: Compute dy/dx using the limit definition of the first derivative for f(x) = 3x2

A: Given function is f(x)=3x2. we have to find the derivative using limit definition .

Q: 2) Use the limit definition to find the derivative of the given function. y = 3r2 – 2x

A: Limit definition of derivative is f'(x)=limh→0f(x+h)-f(x)h

Q: Suppose that f(x, y) is continuous at (2, 3) and that f(2, y) = y3 for y ≠ 3. What is the value f(2,…

A: Given that f(x, y) is continuous at (2, 3) and that f(2, y) = y3 for y ≠ 3. We have to find the…

Q: Prove, using the definition of derivative, that if f(x)  = Cps x,then f'(x) =- Sin x

A: Given fx=cos xBy definition of derivativef'x=limh→0 fx+h-fxhfx=cos xThereforef'x=limh→0 cosx+h-cos…

Q: Use the limit definition to calculate the derivative at x = a of the linear function f(x) = 7.

A:

Q: Let f(x) = x^2 - 4x + 1. What is the equation of the tangent line to f(x) at x = 2? Use the limit…

A: Please see the below picture for detailed solution.

Q: Suppose f(x) = df Find the derivative dx In(In(In(In(x)))). f' (x) :

A:

Q: Consider f(x, y) =(ax² + by²). Based on the second derivative test, what are the conditions…

A: Answer

Q: Find the fourth derivative of f (z) = 37 + In(z) – 164z. Enclose arguments of functions in…

A: To Determine: find the fourth derivative of the function .  Given: we have a function…

Q: 2. Calculate the first derivative of f using the forward and backward difference approach in the…

A: The objective is to calculate the first derivative of f using the forward and backward difference…

Q: Find the derivative of f (x) = 2/x using the limit definition.

A:

Q: Compute the derivative of f (x) = x 1/3 using the limit definition.                            (x +…

A: Given: fx=x13 To Find: The derivative of the given function using limit definition

Q: Determine the derivative of f(x) = ( In x )6 with respect to x. f'(x) =

A:

Q: Use the derivative to find the critical point of f(x) = Ax² + Bx + C. Assume that A, B, and C are…

A:

Q: Find the general anti-derivative of f(x) = + Vx on a continuous interval.

A:

Q: If the function is defined y=2x when x=3, is this function differentiable at x=3? Also, is this…

A:

Q: 2) Let f(x) = x' +x² – 5x – 5. Use the second derivative test to classify the critical points.

A:

Q: Find the critical coordinates of f(x) = x³ – 3x² + 8x + 1. Then, distinguish the local maximum and…

A: Given function is  f(x) = 13x3-3x2+8x+1                 ....(1) Differentiate with respect to x , we…

Q: Find the derivative of f(x) = √x^2 + 1 at a = 2, using the limit definition (i.e. first principles)

A:

Q: Find all the critical points of f(2z, v) = - . 1 + 4x – xy? and classify them using the second…

A: The given function is;  f(x)=-x3/3 +4x -xy2.

Q: Prove that f (x) = x3 − 12x is differentiable. Compute f'(x) and findan equation of the tangent line…

A: For a function f(x) to be differentiable the limit limh→0f(x+h)-f(x)h must exist. This limit is…

Q: Suppose that a firm uses L as an input, and its production function changes with L and a parameter…

A: Let’s apply the above approach to an economic application, namely the profit function of a…

Q: Give an example of a differentiable function ƒ whose first derivative is zero at some point c even…

A:

Q: n(x)=3x?4k+1 %3D

A:

Q: Prove that constant function is differentiable at ever x ER.

A: Since you have asked multiple question, we will solve the first question for you. If you want any…

Q: Let y be a function of x and suppose y=In(x²+y²). What is the correct result for the derivative of y…

A:

Q: Calculate the derivative of the function below using the quotient rule. x4 y = Vx + 2x4 f(x) = and…

A:

Q: b) Use the definition of the derivative to compute f'(x) for 1)f(x) = 2v-5x + 9, 2)f(x) = x sin 3x…

A:

Q: 10. Show that f (x) = { x² sin(1/x) if x + 0 if x is differentiable, but the derivative is not…

A:

Q: Find the derivative of the function f(x) = VI+4 using the limit process.

A: we have to find the derivative

Q: Use the limit definition of the derivative to compute the equation of the tangent line of f (x) =…

A: We have to compute the equation of the tangent line of f (x).

Q: Use the limit definition to calculate the derivative at x = a of the linear function f(x) = 11.

A: Here we have to use limit definition to find derivative

Q: Compute the second derivative of sin(g(x)) at x = 2, assuming that g(2) = 4, 8'(2) = 5, and g"(2) =…

A: Given: The function singx with g2=π4, g'2=5, and g''2=3, To find: The second derivative of given…

Q: 3 Show that f(x)= 1x-5 is NOT differentiable at 5 by using the defini tion of the derivative, (Start…

A:

Q: 3. Use the forward-difference formula to approximate the derivative of f (x)=sin(In x) at x, = 2.5…

A: The derivative of the function can be calculated using the difference method. The formula for the…

Q: .. Consider the quadratic function y = 4-x². Using the limit definition, ind the derivative of the…

A:

Q: Let y be a function of x and suppose y= In(y-x). What is the correct result for the derivative of y…

A:

Q: Use the first derivatives test to find the local exterma of f(x) =x3-12x on R

A:

Q: Use the limit definition of derivative to find the slop of the tangent line of f(x)=5x^3+x at (1,6).

A: The derivative of the function is defined by,

Q: Use backward, forward , and centered difference approximation to estimate the second derivative of…

A:

Q: Show that f (x) = 1x-5| is NOT differentiable at 5 by using the definition of the derivative, (Start…

A:

Q: (b) Show that the function defined by √₂²+y f(x, y) = 0 if (x, y) = (0,0) is not differentiable at…

A: On solving this we will get

Q: 4. By the definition of derivative prove that f(x) = √x² +5 is differen- tiable at x=2.

A:

Q: Use part one of the fundamental theorem of calculus to find the derivative of the function. x. g(>…

A: We know that, the First Fundamental Theorem of Calculus says that if f be a continuous real-valued…

Q: Find the equation of the tangent line of f(x)=2x^2-4x+1 at (2,1) using the limit definition of a…

A:

Q: Find the derivative off with respect to x of f(x) = In x⁹. The derivative off with respect to x of…

A:

Q: 2.) Investigate the derivate of the function f(z)=Imz by using definiton of derivative.

A: The derivative of a function is represented as the change in variable with respect to change in…

Rate of Change

The relation between two quantities which displays how much greater one quantity is than another is called ratio.

Slope

The change in the vertical distances is known as the rise and the change in the horizontal distances is known as the run. So, the rise divided by run is nothing but a slope value. It is calculated with simple algebraic equations as: