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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

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Math 1304 Calculus I

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  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)

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