1 / 8

Math 1304 Calculus I

Math 1304 Calculus I. 3.4 – The Chain Rule. 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

amato
Download Presentation

Math 1304 Calculus I

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  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… f g

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

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

  6. 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

  7. A good working set of rules • Constants: If f(x) = c, then f’(x) = 0 • Powers: If f(x) = xn, then f’(x) = nxn-1 • Exponentials: If f(x) = ax, then f’(x) = (ln a) ax • All 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)

  8. 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) = af(x), then F’(x) = (ln a) af(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) If F(x) = tan(f(x)), then F’(x) = sec2(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)

More Related