Learning Outcomes
- Find the derivative of exponential functions.
- Find the derivative of logarithmic functions.
- Use logarithmic differentiation to determine the derivative of a function.
Textbook
- Chapter 3.9 Derivatives of Exponential and Logarithmic Functions
Textbook Assignment
- p. 331: 331, 334, 337, 340, 341, 346, 347, 351
WeBWork Assignment
- Derivatives-Exponential
- Derivatives-Logarithms
- Derivatives-Logarithmic
Exit problems of the session
-
Find the derivative of the following functions:
(a). $y=3xe^{x^2}$ (b). $y=2^{\sin x\cos x}$ (c). $y=\ln(2x^3-x+1)$ (d). $y=\log(\tan x)$ - Use logarithmic differentiation to find the derivative of the following functions.
(a). $y=(2x)^{\sin x}$ (b). $y=\dfrac{x^2\sqrt{2x+1}}{x^2-1}$
Key Concepts
- Differentiation formulas for the exponential and logarithmic functions.
$\dfrac{d}{dx}(e^x)=e^x$
$\dfrac{d}{dx}(a^x)=a^x \ln a$
$\dfrac{d}{dx}(\ln x)=\dfrac{1}{x}$
$\dfrac{d}{dx}(\log_a x)=\dfrac{1}{x\ln a}$
- Logarithmic differentiation allows us to differentiate functions of the form $y=f(x)^{g(x)}$ or very complex products or quotients by taking the natural logarithm of both sides and exploiting the properties of logarithms before differentiating.
Videos and Practice Problems of Selected Topics
- $\rhd$ Derivative of exponential functions (5:24) Deriving the formula of the derivative of $a^x$.
- $\rhd$ Derivative of logarithmic functions (4:47) Deriving the formula of the derivative of $\log_ax$.
- $\rhd$ An example (3:39) Find the derivative of $7^{x^2-x}$.
- $\rhd$ An example (4:09) Find the derivative of $\log_4(x^2+x)$.
- * Practice: Derivatives of $a^x$ and $\log_ax$. (4 problems)
