Calculus Review: Derivatives and Equation of Tangent and Normals

What is the product rule for derivatives?

• If @@u(x)@@ and @@v(x)@@ are functions, then @@ (uv)' = u'v + uv' @@.

• If @@u(x)@@ and @@v(x)@@ are functions, then @@ (uv)' = u' + v' @@.

• If @@u(x)@@ and @@v(x)@@ are functions, then @@ (uv)' = uv' @@.

• If @@u(x)@@ and @@v(x)@@ are functions, then @@ (uv)' = u + v @@.

What is the derivative of the function @@f(x) = x^4 \, cos(x)@@?

• @@f'(x) = x^4 \, cos(x) + 4x^3 \, sin(x)@@

• @@f'(x) = 4x^3 \, cos(x) - x^4 \, sin(x)@@

• @@f'(x) = 4x^3 \, cos(x) + x^4 \, sin(x)@@

• @@f'(x) = 4x^3 \, cos(x) + 4x^4 \, sin(x)@@

What is the definition of a derivative?

• The derivative of a function at a point is the limit of the average rate of change of the function as the interval approaches zero.

• The derivative measures the total area under the curve of a function.

• The derivative is the slope of the tangent line to the function at a given point.

• The derivative represents the maximum value of a function over an interval.

What is the significance of the normal line in relation to the tangent line?

• The normal line is parallel to the tangent line at the point of tangency.

• The normal line is perpendicular to the tangent line at the point of tangency.

• The normal line intersects the tangent line at a 45-degree angle.

• The normal line is the same as the tangent line.

How do you find the slope of the tangent line at a point?

• By calculating the area under the curve at that point.

• By taking the second derivative of the function.

• By evaluating the derivative of the function at that point.

• By finding the average rate of change over an interval.

What is the derivative of @@f(x) = e^x@@?

• @@f'(x) = e^x@@

• @@f'(x) = e^x + 1@@

• @@f'(x) = xe^{x-1}@@

• @@f'(x) = 0@@

• @@y' = (3x^4 - 7)(10x) + (12x^3)(5x^2 + 1)@@

• @@y' = (12x^3)(5x^2 + 1)@@

• @@y' = (15x^2)(3x^4 - 7)@@

• @@y' = (3x^4)(10x) - (7)(5x^2)@@

What is the quotient rule for derivatives?

• If @@u(x)@@ and @@v(x)@@ are functions, then @@(u/v)' = (u'v - uv')/v^2@@.

• If @@u(x)@@ and @@v(x)@@ are functions, then @@(u/v)' = (uv' - u'v)/u^2@@.

• If @@u(x)@@ and @@v(x)@@ are functions, then @@(u/v)' = (u' + v')/v@@.

• If @@u(x)@@ and @@v(x)@@ are functions, then @@(u/v)' = u'v + uv'@@.

What is the formula for the equation of a line given a point and a slope?

• @@y - y_1 = m(x - x_1)@@

• @@y = mx + b@@

• @@y - y_1 = b(x - x_1)@@

• @@y = m(x - y_1) + y_1@@