Chain Rule Differentiation

The Chain Rule is a formula for computing the derivative of the composition of two or more functions. It states that if you have a function y = f(g(x)), then the derivative y' = f'(g(x)) * g'(x).

f'(x) = 5(x^2 + 2x)^4(2x + 2)

To find the equation of the tangent line at a point (a, f(a)), calculate the derivative f'(a) to get the slope, then use the point-slope form: y - f(a) = f'(a)(x - a).

y = 500x - 400

f'(x) = 105(3x + 4)^4

f'(x) = 6x^5 - 16x^3 + 8x

f'(x) = 30x^5(x^6 + 4)^4

The power rule states that if f(x) = x^n, then f'(x) = n*x^(n-1), where n is a constant.

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

The quotient rule states that if you have a function that is the quotient of two functions u(x) and v(x), then the derivative is (u'v - uv')/v^2.