Differential Calculus Exam Questions with Correct Grade A+ Solutions

Differential Calculus Exam Questions with Correct Grade A+ Solutions First Derivative - Correct Answers: f(x)=3x³+2x²-x f'(x)= 9x²+4x-1 Chain Rule - Correct Answers: f'(x)= n[f(x)]ⁿ⁻¹f'(x) Product Rule - Correct Answers: f'(x)= f'(x)g(x)+f(x)g'(x) Quotient Rule - Correct Answers: f'(x)= (f'(x)g(x)-f(x)g'(x))/[g(x)]² Implicit Differentiation - Correct Answers: f'(x)= nyⁿ⁻¹(dy/dx) Tangents - Correct Answers: 1) Find the slope (dy/dx) 2) sub in the x co-ordinate to the slope 3) sub in the x co-ordinate to the equation to find the y 4) sub all into y=mx+c 5) Solve for c Normals - Correct Answers: 1) Find the slope (dy/dx) 2) sub in the x co-ordinate to the slope 3) then find the negative reciprocal of the slope 4) sub in the x co-ordinate to the equation to find the y 5) sub all into y=mx+c 6) Solve for c Second Derivative - Correct Answers: y=4x³+4x²+x-1 dy/dx=12x²+8x+1 d²y/dx²=24x+8 First principles - Correct Answers: f'(x)=lim/h→0 (f(x+h)-f(x))/h Rearrange, expand, cancel h's, then h=0 ∴ derivative The Derivative function of y=e(x) - Correct Answers: e(x) = e(x) e(f(x)) = e(f(x))f'(x) ke(f(x)) = ke(f(x))f'(x) Natural Logarithmic functions - Correct Answers: lnA+lnB = lnAB lnA - lnB = (lnA)/(lnB) lnAⁿ = nlnA ln(1/A) = lnA⁻¹ = -lnA e(x) = a ∴ x = ln a ∴e(lna) = a ln 1 =0 lne = 1 lneⁿ = n Solving Exponential Equations - Correct Answers: log both sides dont log both sides if more than 1 term use log laws to get two terms then log both sides and solve IF e(-x) MULTIPLY BY e(x) The derivative of natural logarithmic functions - Correct Answers: ln x = 1/x ln f(x) = f'(x)/f(x) k ln f(x) = k(f'(x)/f(x)) ln y = 1/y (dy/dx)