Derivatives of Exponential and Logarithmic Functions: Fresh Take

  • Determine the derivatives of exponential and logarithmic functions
  • Apply logarithmic differentiation to find derivatives

Derivative of the Exponential Function

The Main Idea 

  • The Natural Exponential Function:
    • Defined as E(x)=exE(x)=ex
    • e2.718281828...e2.718281828...
  • Key Property:
    • ddx(ex)=exddx(ex)=ex
  • General Exponential Function:
    • For B(x)=bxB(x)=bx where b>0b>0: B(x)=bxB(0)B(x)=bxB(0)
  • Chain Rule Application:
    • ddx(eg(x))=eg(x)g(x)ddx(eg(x))=eg(x)g(x)

Find the derivative of h(x)=xe2xh(x)=xe2x.

Find the derivative of f(x)=etan(2x)f(x)=etan(2x).

If A(t)=1000e0.3t describes the mosquito population after t days, as in the preceding example, what is the rate of change of A(t) after 4 days?

Derivative of the Logarithmic Function

The Main Idea 

  • Derivative of Natural Logarithm:
    • For x>0, if y=lnx, then dydx=1x
  • General Logarithmic Derivative:
    • For h(x)=ln(g(x)), h(x)=g(x)g(x)
  • Derivative of General Base Logarithm:
    • For b>0,b1, if y=logbx, then dydx=1xlnb
  • Derivative of General Exponential:
    • For b>0,b1, if y=bx, then dydx=bxlnb

Differentiate: f(x)=ln(3x+2)5.

Find the derivative of f(x)=ln(x2sinx2x+1).

Find the slope of the tangent line to y=log2(3x+1) at x=1.

Find the slope for the line tangent to y=3x at x=2.

Logarithmic Differentiation

The Main Idea 

  • Purpose:
    • Simplify differentiation of complex functions, especially those involving products, quotients, and exponents
  • Applications:
    • Functions of the form y=(g(x))n
    • Functions like y=bg(x)
    • Expressions such as y=xx or y=xπ

Step-by-Step Process

  1. Take natural logarithm of both sides: lny=ln(h(x))
  2. Expand using logarithm properties
  3. Differentiate both sides implicitly
  4. Solve for dydx
  5. Simplify the final expression

Key Logarithm Properties

  • Product Rule: logb(MN)=logb(M)+logb(N)
  • Quotient Rule: logb(MN)=logb(M)logb(N)
  • Power Rule: logb(Mn)=nlogb(M)
  • Change of Base: logb(M)=logn(M)logn(b)

Find the derivative of y=x2x+1exsin3x

Use logarithmic differentiation to find the derivative of y=xx.

Find the derivative of y=(tanx)π.

Find the derivative of y=(2x4+1)tanx.

Find the derivative of y=xr where r is any real number.