Derivatives of Exponential and Logarithmic Functions: Learn It 3

Logarithmic Differentiation

We now explore derivatives of functions like y=(g(x))n for certain values of n, and those involving forms like y=bg(x), where b>0 and b1. These functions present a challenge when trying to find derivatives directly. To address this, we employ logarithmic differentiation. This technique is particularly useful for differentiating functions of the form y=xx or y=xπ.

Logarithmic differentiation simplifies the process by taking the natural logarithm of both sides, which transforms multiplicative relationships into additive ones, making the derivative more straightforward to compute. We will apply this technique to solve problems such as finding the derivate of y=x2x+1exsin3x.

We outline this technique in the following problem-solving strategy.

How To: Use Logarithmic Differentiation

  1. Begin by Logarithmizing: To apply logarithmic differentiation to the function y=h(x), start by taking the natural logarithm of both sides of the equation, giving lny=ln(h(x)).
  2. Expand Using Logarithm Properties: Utilize the properties of logarithms to simplify ln(h(x)) as much as possible. This may involve using the logarithm rules for multiplication, division, and powers.
  3. Differentiate Both Sides: With y implicitly defined by the equation lny=lnh(x), differentiate both sides with respect to x. On the left, apply the chain rule to obtain 1ydydx, and on the right, use the derivative of lnh(x).
  4. Isolate dydx Multiply through by y to solve for dydx
  5. Simplify the Derivative: Simplify the expression for dydx to obtain the final derivative in terms of x.

It may be useful to review your properties of logarithms. These will help us in step 2 to expand our logarithmic function.

The Product Rule for Logarithms logb(MN)=logb(M)+logb(N)
The Quotient Rule for Logarithms logb(MN)=logbMlogbN
The Power Rule for Logarithms logb(Mn)=nlogbM
The Change-of-Base Formula logbM=lognMlognb n>0,n1,b1

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

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