1. Math 1304 Calculus I 3.4 – The Chain Rule

2. Ways of Stating The Chain Rule Statements of chain rule: If F = fog is a composite, defined by F(x) = f(g(x)) then F'(x) = f'(g(x))g'(x) If y = f(u) and u = g(x) are differentiable, then dy/du = dy/dx dx/du Other ways of writing it (fog)'(x) = f'(g(x)) g'(x) Basic ideas - for a chain of functions, rates multiply together

3. The Chain Rule The derivative of the composition is… g f

4. The Chain Rule The derivative of the composition is the product of the derivatives g f f  g (f  g)’=f’  g’ g’ f’ z x y

5. The Chain Rule The derivative of the composition is the product of the derivatives g f f  g (f  g)’=f’  g’ g’ f’ z x y

6. Proof of Chain Rule Use the notation dy/dx, show that if y=g(x) and z=f(y), then dz/dx = dz/dy dy/dx

7. Basic Approach to Chain Rule Identify inside and outside functions Take the derivative of outside function (put inside function inside, as is) Multiply by derivative of inside function

8. A good working set of rules Constants: If f(x) = c, then f’(x) = 0 Powers: If f(x) = x n, then f’(x) = nx n-1 Exponentials: If f(x) = a x, then f’(x) = (ln a) a x Trigonometric functions: If f(x) = sin(x), then f’(x) = cos(x) If f(x) = cos(x), then f’(x) = - sin(x) Scalar mult: If f(x) = c g(x), then f’(x) = c g’(x) Sum: If f(x) = g(x) + h(x), then f’(x) = g’(x) + h’(x) Difference: If f(x) = g(x) - h(x), then f’(x) = g’(x) - h’(x) Multiple sums: derivative of sum is sum of derivatives Linear combinations: derivative of linear combo is linear combo of derivatives Product: If f(x) = g(x) h(x), then f’(x) = g’(x) h(x) + g(x)h’(x) Multiple products: If f(x) = g(x) h(x) k(x), then f’(x) = g’(x) h(x) k(x) + g(x) h’(x) k(x) + g(x) h(x) k’(x) Quotient: If f(x) = g(x)/h(x), then f’(x) = (g’(x) h(x) - g(x)h’(x))/(h(x)) 2 Composition: If F = fog is a composite, defined by F(x) = f(g(x)) then F'(x) = f'(g(x))g'(x)

9. A new good working set of rules Constants: If F(x) = c, then f’(x) = 0 Powers: If F(x) = f(x) n, then F’(x) = n f(x) n-1 f’(x) Exponentials: If F(x) = a f(x), then F’(x) = (ln a) a f(x) f’(x) All trigonometric functions: If F(x) = sin(f(x)), then F’(x) = cos(f(x)) f’(x) If F(x) = cos(f(x)), then F’(x) = - sin(f(x)) f’(x) Scalar mult: If F(x) = c f(x), then F’(x) = c f’(x) Sum: If F(x) = g(x) + h(x), then F’(x) = g’(x) + h’(x) Difference: If F(x) = g(x) - h(x), then F’(x) = g’(x) - h’(x) Multiple sums: derivative of sum is sum of derivatives Linear combinations: derivative of linear combo is linear combo of derivatives Product: If F(x) = g(x) h(x), then F’(x) = g’(x) h(x) + g(x)h’(x) Multiple products: If F(x) = g(x) h(x) k(x), then F’(x) = g’(x) h(x) k(x) + g(x) h’(x) k(x) + g(x) h(x) k’(x) Quotient: If F(x) = g(x)/h(x), then F’(x) = (g’(x) h(x) - g(x)h’(x))/(h(x)) 2 Composition: If F = fog is a composite, defined by F(x) = f(g(x)) then F'(x) = f'(g(x))g'(x)