Ch-3 Notes_Sec 3.4&3.5

Chapter 3: Calculus for Pharmacy and Allied Health

Section 3.4: The Product Rule, Quotient Rule, and Trigonometric Functions

Differentiation Formulas

• ( \frac{d}{dx}(c) = 0 )

• ( \frac{d}{dx}(x^n) = nx^{n-1} )

• ( \frac{d}{dx}(e^x) = e^x )

• ( \frac{d}{dx}(cf) = c\frac{d}{dx}(f) )

• ( \frac{d}{dx}(f + g) = \frac{d}{dx}(f) + \frac{d}{dx}(g) )

• ( \frac{d}{dx}(f - g) = \frac{d}{dx}(f) - \frac{d}{dx}(g) )

• ( \frac{d}{dx}(fg) = f\frac{d}{dx}(g) + g\frac{d}{dx}(f) )

• ( \frac{d}{dx}(\frac{f}{g}) = \frac{g\frac{d}{dx}(f) - f\frac{d}{dx}(g)}{g^2} )

Examples of Differentiation

1. Differentiate the following:

   • (a) ( y = x^3 \sin(x) )

   • (b) ( f(t) = \sqrt{t} (a + bt) )

   • (c) ( f(x) = (-2x^2 + 9x)(7e^x + 6) )

2. Finding Derivatives with Given Conditions:

   • If ( h(x) = xg(x) ) and ( g(3) = 5, \ g'(3) = 2 ), find ( h'(3) )

3. Differentiate to find derivatives:

   • (a) ( y = e^{x^2}(1+x) )

4. Tangent Line Equations:

   • Find an equation of the tangent line to ( y = e^x(1+x^3) ) at the point ( (1, e^1) )

Derivatives of Trigonometric Functions

• ( (\sin x)' = \cos x )

• ( (\csc x)' = -\csc x \cot x )

• ( (\cos x)' = -\sin x )

• ( (\sec x)' = \sec x \tan x )

• ( (\tan x)' = \sec^2 x )

• ( (\cot x)' = -\csc^2 x )

Examples in Classwork

1. Differentiate the following:

   • (a) ( g(x) = \sqrt{x} e^x )

   • (b) ( h(t) = e^t \sin(t) )

   • (c) ( f(x) = (x + e^x)^3 - \sqrt{x} )

   • Etc.

Chapter 3: Section 3.5: The Chain Rule and Implicit Differentiation

The Chain Rule

• Definition: To find the derivative of a composite function ( F = f \circ g = f(g(x)) ).

• Formulation:

  • If ( g ) is differentiable at ( x ) and ( f ) is differentiable at ( g(x) ), then:

  • ( F'(x) = f'(g(x)) , g'(x) )

Chain Rule with Power Rule

• If ( n ) is any real number and ( u = g(x) ) is differentiable, then:

  • ( \frac{d}{dx}[u^n] = nu^{n-1} , u' )

Examples Using Chain Rule

1. Find ( f'(x) ) for:

   • (a) ( f(x) = \sqrt{x^2 + 1} )

   • (b) ( f(x) = (2x + 5)^3 )

Implicit Differentiation

Definition and Procedure

• Implicit Differentiation: To differentiate equations where y is not explicitly solved for x.

• Steps:

  1. Differentiate both sides concerning x.

  2. Apply the rules of differentiation.

  3. Group terms containing ( y' ).

  4. Solve for ( y' ).

Example of Implicit Differentiation

• Given ( x^3 + y^3 = 6xy )

• Steps:

  1. Differentiate: ( 3x^2 + 3y^2 y' = 6y + 6xy' )

  2. Isolate and solve for ( y' ) after substituting values.

Related Rates

Understanding Related Rates

• Related rates problems involve finding how the rate of change of one quantity relates to the rate of change of another.

• Start with a relation equation between quantities and then differentiate with respect to time.

Example Problem

• When the diameter of a spherical tumor is 16 mm and grows at a rate of 0.4 mm/day, find how fast the volume is changing at that time.

• Volume formula: ( V = \frac{4}{3} \pi r^3 )

• Differentiate to find ( \frac{dV}{dt} ).

Conclusion

• Engage in practice exercises and examples to reinforce understanding of differentiation and applications in the medical context.