Derivatives of Exponential Functions TS Explicitly Assessing Information

Derivatives of Exponential Functions TS: Explicitly Assessing Information and Drawing Conclusions

Objective To find the derivatives of natural exponential functions.

Derivatives of Exponential Functions What is the derivative? Super Guess:

Derivatives of Exponential Functions What is the derivative? Super Guess: This is incorrect. This rule works for f (x) = x. N, when the base is a variable x and the exponent is a constant N. The power rule does not apply to exponential functions.

Derivatives of Exponential Functions What is the derivative?

Derivatives of Exponential Functions Use the chain rule to find the derivative of the composition of the natural exponential function and another function.

Derivatives of Exponential Functions The derivative of an exponential function of the form f (x) = eu is the product of the function and the derivative of its exponent.

Derivatives of Exponential Functions Derivative of eu is eu Differentiate: Derivative of 5 x is 5

Derivatives of Exponential Functions Differentiate: Derivative of eu is eu Derivative of x 3 is 3 x 2

Derivatives of Exponential Functions Differentiate: Derivative of eu is eu Derivative of 3 x 2 is 6 x

Derivatives of Exponential Functions Differentiate: Derivative of eu is eu Derivative of x-3 is -3 x-4

Derivatives of Exponential Functions Differentiate: Product Rule

Derivatives of Exponential Functions Quotient Rule Differentiate:

Derivatives of Exponential Functions Differentiate:

Derivatives of Exponential Functions Differentiate:

Derivatives of Exponential Functions Product Rule Differentiate:

Derivatives of Exponential Functions Quotient Rule Differentiate:

Conclusion The derivative of the natural exponential function is itself. The derivative of an exponential function of the form f (x) = eu is the product of the function and the derivative of its exponent.