Derivatives on Product & Quotient Rule

Front

What is the Product Rule for derivatives?

Back

If u(x) and v(x) are functions, then the derivative of their product is given by: (u*v)' = u'v + uv'.

Front

What is the Quotient Rule for derivatives?

Back

If u(x) and v(x) are functions, then the derivative of their quotient is given by: (u/v)' = (u'v - uv')/v^2.

Front

Find the derivative of g(x) = (3x - 2)/(x^2 + 2).

Back

g'(x) = (-3x^2 + 4x + 6)/(x^2 + 2)^2.

Front

Find the derivative of the function f(x) = x^3 + x^2 + 12.

Back

f'(x) = 3x^2 + 2x.

Front

What does it mean for the slope of the tangent line to be zero?

Back

The slope of the tangent line is zero at points where the function has a local maximum or minimum.

Front

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

Back

Use the point-slope form: y - f(a) = f'(a)(x - a), where a is the point of tangency.

Front

Find the tangent line to the curve y = \frac{12}{x^2} at x = 2.

Back

The tangent line is y = -3x + 9.

Front

What is the significance of the derivative in calculus?

Back

The derivative represents the rate of change of a function with respect to its variable.

Front

How do you determine critical points of a function?

Back

Critical points occur where the derivative is zero or undefined.

Front

What is the relationship between the first derivative and increasing/decreasing functions?

Back

If f'(x) > 0, the function is increasing; if f'(x) < 0, the function is decreasing.