Exam 2 - Math 2413 Calculus 1 Questions with Complete Solutions

Constant Rule - ANSWER-d/dx [c] = 0 Constant Multiple Rule - ANSWER-d/dx [cf(x)]=cf'(x) Sum Rule - ANSWER-d/dx [f(x)+g(x)] = f'(x)+g'(x) Difference Rule - ANSWER-d/dx [f(x)-g(x)] = f'(x)-g'(x) Product Rule - ANSWER-d/dx [f(x)*g(x)] = f(x)g'(x) + f'(x)g(x) Quotient Rule - ANSWER-d/dx [f(x)/g(x)] = [g(x)f'(x) - g'(x)f(x)]/[g(x)²] Chain Rule - ANSWER-d/dx [f(g(x)] = f'(g(x))*g'(x) Power Rule - ANSWER-d/dx[xⁿ]=nxⁿ⁻¹ Definition of the number e - ANSWER-d/dx[eⁿ] = eⁿ (n = x) Exponential Rule (any base) - ANSWER-d/dx[bⁿ] = bⁿ ln b (n = x) Natural Log Rule - ANSWER-d/dx ln |x| = 1/x Logarithmic Differentiation - ANSWER-d/dx (logₙx) = 1/x ln b (n = b) Derivative of Sine - ANSWER-cos(x) Derivative of Cosine - ANSWER--sin(x) Derivative of Tangent - ANSWER-sec²x Derivative of Cotangent - ANSWER--csc²x Derivative of Cosecant - ANSWER--csc(x)cot(x) Derivative of Secant - ANSWER-sec(x)tan(x) Derivative of Inverse Sine - ANSWER-1/√1-x² Derivative of Inverse Cosine - ANSWER--1/√1-x² Derivative of Inverse Tangent - ANSWER-1/1+x²