Derivatives Review (Oct 17)

The derivative is $$f'(x) = 3x^2 + 2x$$ .

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

The first step is to apply the product rule: $$2x^2(3) + (3x - 4)(4x)$$ .

The notation 'dy' is not a correct notation for 'derivative'.

The slopes will be zero at points B and D.

The derivative of a function measures how the function's output value changes as its input value changes.

The product rule states that if you have two functions $$u(x)$$ and $$v(x)$$ , then the derivative of their product is $$(uv)' = u'v + uv'$$ .

A function is differentiable at a point if it has a derivative at that point, meaning it has a defined slope of the tangent line.

A zero derivative indicates that the function has a horizontal tangent line at that point, which may correspond to a local maximum or minimum.

To find the derivative of a polynomial function, apply the power rule: $$f'(x) = n*x^{n-1}$$ for each term $$ax^n$$ .