Let r(t) be the rate at which the world’s oil is consumed,where is measured in years starting at t =0 on January 1,2000, and r(t) is measured in barrels per year. What does ∫03  r(t) dtrepresent?

Q: Find the derivative of the function 1-x2 t2sin(t) dt g'(x) =

A: First I have written the Leibnitz rule for derivative of an integral and using them find the…

Q: B) Find the third derivative of the function: Y = + VX3 %3D

A: We have to find third derivative

Q: Find the derivative of the function.

A: Given, To evaluate derivative

Q: Calculate the derivative e12x d In (t)dt dx using Part 2 of the Fundamental Theorem of Calculus.

A:

Q: dx y=u(1+u°)³ and u =(3x-2)². %3D Express the derivative in terms of x, if

A:

Q: F(2) = [*e*ax e' dt

A: We have to find derivative

Q: Find the derivative of the function at P in the direction of A g(x, y) = x - (v² /x)+ /3 sec*(2xy),…

A: To find: The derivative of the function at P0=1,1 in the direction of  A=12i+5j Given information:…

Q: Find the derivative of the function. x² F(x) e dt %3D F'(x) =

A: The fundamental theorem of Calculus says: If F(x)=∫h(x)g(x)f(t)dt⇒F'(x)=f(g)g'-f(h)h' Here,…

Q: Let f(x, y) = sin(3x + 5y) and P = 3 Find the maximum rate of change of the fucntion f at the point…

A: Find gradient of f(x,y). Find compute the point . Take the magnitude of the same .

Q: d dx 156. dt 1+14 4°

A:

Q: What is the derivative of y = V(x? V(a2 – 1) (x + 1) ? -

A:

Q: Find the third derivative. 1 (x + y)Z + (x – y)Z = a

A:

Q: B) F(x) = "* In x sin e'dt

A:

Q: Find the derivative of the function f(t) = te-t. f'(t) =

A:

Q: (b) Find and simplify the derivative of the function pln(z) e dt. f(x) = |

A: we need to find the derivative of given definite integral

Q: Find the linear approximation of f(x) = ln x at x = 1 and use it to estimate ln(1.33).

A: The linear approximation of f(x) at x=a is given by L(x)=f(a)+f'(a)(x−a) Given, f(x)=ln(x) at x=1.…

Q: d Use the Fundamental Theorem of Calculus to find the derivative. dx dt 1+ vE

A:

Q: Calculate the derivative of t2 -k 3 - t F(t) = cos(t) i + 3e2t+2j + |

A:

Q: Find the derivative of the function at Po in the direction of A. g(x, y) = x – (y²/x) + V3 sec-1…

A:

Q: Find the derivative of the function. s=? In |t| ds dt II

A:

Q: Find the derivative of the function.

A: Given function is, F(x)=∫0e2xln(t+1)dt

Q: Find the interval in which function f(x)=sin (log,x)+cos(log,x) is decreasing

A:

Q: (b) Find and simplify the derivative of the function cln(r) f(r) = dt. %3D

A: Using Leibniz integral rule for differentiation

Q: Use the Fundamental Theorem of Calculus to find the derivative. Vx-x1-t2 dt d dx

A: The Fundamental Theorem of Calculus says: ddx∫ag(x)f(t) dt=f(g(x))·g'(x)...(1)

Q: Find the derivative of t2 + e3t e(t²+1) f(t) = %3D

A:

Q: find the second derivative of y = x In x A 1/x в) 1/2х D) 2x

A:

Q: Find the derivative of the function at Po in the direction of A. g(x, y) = x - (y/x) + V3 sec-…

A: Given problem:-

Q: Find the derivative of the function.

A: Given: y=∫2x3x+1sint4dt

Q: Find the 127th derivative of (1 t a*) 16 at r 0.

A: we need to find 127 derivative of given f(x) so we need to apply product rule

Q: Find the derivative of the function. h(t) = 1 2 dt

A: We have to find the derivative of the given function.

Q: Find the derivative of the function. g(t) = ln t/t2

A:

Q: Calculate the derivative d dx t14 + 5dt using Part 2 of the Fundamental Theorem of Calculus.

A:

Q: Find the derivative of the function. h(x) = log,(x² + x) h'(x) %3D

A: Given query is to find the derivative of the expression.

Q: 3. Find the derivative of f(x, y, z) = z2 + ye" at (In 2, 2, 1/2) in the direction in which the…

A:

Q: If f(t) is measured in dollars per year and t is measured in years, what are the units of f(t)dt?

A: We will find out the required unit.

Q: Find the derivative df/dt. * f (t) = 2' + In (t)

A: Given f(t)= 2t+lnt

Q: Compute the derivative of x(x + 2) (2x + 1)(3x + 2) y = (a) directly using the Chain and Product…

A:

Q: Find the derivative of the function In t dt.

A:

Q: Find the derivative of function ei, x#0 1 еx, f (x) = 0. at xo = 0 X = 0

A: Find the derivative of the given function.   Given: fx=e1x, x≠00, x=0

Q: Find the derivative of the function.

A: Given- gx=∫1cosx1-t23dt To find- The derivative of the above function. Concept Used- Using the…

Q: Find the derivative of g(x) = |. •r In z Vt2 +t+1 dt

A:

Q: 2 2 X = 1+t2 y=r(1+t?)

A: To find second derivative of given function.

Q: Show that, for a function f(x) with anti-derivative F(x), d s)ds f(x). %3D dx a

A: Suppose f(x) has an anti-derivate F(x), v(x) and u(x) are differentiable, then Leibniz's rule of…

Q: Find the derivative of the function. y = In) x +6. (a)"* g(x) = log,(x) 9'(x) = (b) y = x5x + 2 (c)

A:

Q: Use differentials to approximate the change in f(x)=x3+x2-2x+1 when x=2 and dx=0.05 a. 0.1 b. 7…

A: Use differentials to approximate the change in f(x)=x3+x2-2x+1 when x=2 and dx=0.05 a. 0.1 b. 7 c.…

Q: The derivative of f(z) = Log(2iz - 4+6i) at z = -3+i is

A:

Q: In Exercises 7–12, use the Quotient Rule to calculate the derivative.

A: gt=t2+1t2-1 We need to find the derivative of the function at t=-2

Q: Calculate the derivative of the function g(x)=x∫√x(t2−t)dt at x=1.

A: using the fundamental theorem of calculus we find g'(x)

Q: Find the third derivative of f (z) = T2x +I. fll! (z) =

A: Explanation of the answer is as follows