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Activity 5 5 Derivatives and Antiderivatives of Exponentials and

Activity 5 5 Derivatives and Antiderivatives of Exponentials and 2025

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The document focuses on the derivatives and antiderivatives of exponential and logarithmic functions. It includes discussions on derivative formulas for natural and base b logarithms, as well as exponential functions. The document contains exercises for computing derivatives, evaluating integrals, and applying differentiation rules, including logarithmic differentiation. Additionally, it touches on thermodynamic concepts related to enthalpy and entropy.

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What is the derivative of e to the power 3?

0:17 0:48 And the derivative of a constant no matter what it is is always zero. So the answer here is actuallyMoreAnd the derivative of a constant no matter what it is is always zero. So the answer here is actually just the derivative of e cubed is equal to zero.

What is the derivative of e to the 4x power?

0:20 1:09 We need to consider the derivative with respect to U. And then we have d u d x. This is obtained byMoreWe need to consider the derivative with respect to U. And then we have d u d x. This is obtained by the chain rule of derivatives. Now differentiating we will get e to the power U times 4.

What is the derivative of e to a power?

Integration of a Logarithmic Form Thus, reversing the process where the denominators exponent is would lead to an integral of the logarithmic form. Thus, the integration of x d x 3 + x 2 will be done using the logarithmic form, whereas x d x ( 3 + x 2 ) 2 will be done using the power rule for integration.

How do you find the antiderivative of an exponential function?

The differentiation of e to the power x is equal to e to the power x because the derivative of an exponential function with base e is equal to ex. Mathematically, it is denoted as d(ex)/dx = ex. e to the power x is an exponential function with a base equal to e, which is known as Eulers number.