- 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
- e≈2.718281828...e≈2.718281828...
- 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,b≠1, if y=logbx, then dydx=1xlnb
- Derivative of General Exponential:
- For b>0,b≠1, 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
- Take natural logarithm of both sides: lny=ln(h(x))
- Expand using logarithm properties
- Differentiate both sides implicitly
- Solve for dydx
- 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=x√2x+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.