Find f^'(x) by Formula (7) and then by logarithmic differentiation. f(x)=π^sinx

Name: Problem 29, Chapter 3: Calculus Early Transcendentals
Uploaded: 2020-08-26T09:53:17-08:00
Duration: 1 min 29 s
Channel: Zachary Mitchell
Description: Find f^'(x) by Formula (7) and then by logarithmic differentiation. f(x)=π^sinx

step 1 So first we're going to go ahead and use formula 7 in order to differentiate. And then we're als

Find f^'(x) by Formula (7) and then by logarithmic...

Key Concepts

Differentiation of Exponential Functions

When dealing with functions of the form a^(g(x)), where a is a positive constant and g(x) is an arbitrary differentiable function, the derivative is found by multiplying the original function by the natural logarithm of the base and the derivative of the exponent. This method provides a direct application of the general formula d/dx[a^(g(x))] = a^(g(x)) * ln(a) * g'(x), illustrating how the constant base’s logarithm scales the derivative of the exponent.

Logarithmic Differentiation

Logarithmic differentiation is a technique used to simplify the differentiation of functions where the variable appears in the exponent. By taking the natural logarithm of both sides of an equation, exponentiation is transformed into multiplication, which often makes the differentiation process more straightforward. This method is particularly helpful when the function is complex or when both the base and the exponent are functions of x, although in this case the base is constant.

Chain Rule

The chain rule is a fundamental principle in calculus used for differentiating composite functions. It states that the derivative of a composite function is the derivative of the outer function evaluated at the inner function, multiplied by the derivative of the inner function. This is essential when differentiating functions where one function is nested within another, such as in the exponent of an exponential function, ensuring that all dependencies on the variable are appropriately accounted for.

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