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- A. Use implicit differentiation to find the derivative: sin(xy)=x6 - y B. Find the equation of the line normal to x3y-2xy2=-4 at the point (-2,1)Find an equation of the tangent line to the graph of the function f through the point (x0, y0) not on the graph. To find the point of tangency (x, y) on the graph of f, solve the equation. f′(x) = (y0 − y)/(x0 − x) f(x) = √x, (x0, y0) = (−4, 0)Anti-derivative of this function: h(θ) = 4 csc θ cot θ + 2 sec2 θ
- Derive Newton's result that the fluxion of an arc to the fluxion of its secant is ats its cosine to its secant. Use the result on the tangent already demonstrated and the fluxional relationship derived CT^2+AT^2+AC^2, noting that AC, the radius of the circle, is fixed. Translate this result into a standard modern result on the derivative of the secant. See this image:Find the derivative of the function. Simplify and express the answer using positive exponents only. y = (x2 − 3)3 x2 + 3 Step 1 Recall the Quotient Rule: If f(x) = u(x) v(x) where u and v are differentiable functions of x, with v(x) ≠ 0, then f '(x) = v(x) · u'(x) − u(x) · v'(x) v(x) 2 . We can apply the Quotient Rule to find the derivative of y = (x2 − 3)3 x2 + 3 if we let u(x) = (x2 − 3)3 and v(x) = x2 + 3. Begin by finding the derivative of v(x). If v(x) = x2 + 3, then v'(x) =Consider the function f(x) = arctan2(1 + arctan6(1 + arctan3(x))). Use the formulas f0’≈ f1-f-1/2h andf0”≈ f-1-2f0+f1/h2 for the numerical differentiation with h = 0:00001 to approximate the value f’(π/3) of the first and the value f”(π/3) of the second derivative of f at x0 = π/3. Show your work by creating a table with the values of the function f(x) of the form k xk fk ... ... ... ... ... ... ... ... ... All calculations are to be carried out in the FPA9.
- Consider the function f(x)=sinxcosx on the interval 0≤x≤π. (a) Calculate the derivative of f. (b) Find all numbers on the interval [0,π] where f has a horizontal tangent line. (c) Calculate the second derivative of f. (d) Find all numbers in the interval [0,π] where f″(x)=0.3 -))F (x, y) = (cos x). (Sin y) - (cos y). Which of the following is the derivative of the function (sin x)? A) cosx + sinyB) cos (x + y)C) 1D) sin (x.y)E) cos²x + sin²y = 1A cardboard box manufacturer wishes to make open boxes from rectangular pieces of cardboard with dimensions 16cm by 30cm by cutting equal squares from the four corners and turning up the sides. What is the domain of V? Find the derivative of V(x) with respect to x.
- a. Use the product rule to find the derivative of the given function. b. Find the derivative by expanding the product first. h(z)=3−z2z3−3z+4Consider the fuction f(x) = sin x cos x on the interval 0 ≤ x ≤ π. (a) Calculate the derivative of f. (b) Find all numbers of the interval [0, π] where f has a horizontal tangent line. (c) Calculate the second derivative of f. (d) Find all numbers of the interval [0, π] where f " (x) = 0.Find a linearization that will replace the function over an interval that includes the given point X 0. Center the linearization not at X0 ,but at a nearby integer, x=a, at which the given function and its derivative are easy to evaluate. f(x)= 3^square root x , X0=26.7