Αποτελέσματα Αναζήτησης
20 Νοε 2021 · Let's see what happens when we try to compute the derivative of this function just using the definition of the derivative.
25 Ιουλ 2021 · Find the derivative of \[e^{2x}.\nonumber \] Solution. We use the chain rule with \[y = e^u, \;\; u = 2x.\nonumber \] Which gives \[y'=e^u, \;\; u'=2.\nonumber \] So that \[(e^{2x})'=(e^u)(2)=2e^{2x}.\nonumber \]
Learning Objectives. 3.9.1 Find the derivative of exponential functions. 3.9.2 Find the derivative of logarithmic functions. 3.9.3 Use logarithmic differentiation to determine the derivative of a function. So far, we have learned how to differentiate a variety of functions, including trigonometric, inverse, and implicit functions.
21 Ιουν 2023 · Using the definition of the derivative, calculate the derivative of the function \(y=a^{x}\) for an arbitrary base \(a>0\). Describe the significance of the special base \(e\). Summarize the properties of the function \(e^{x}\), its derivatives, and how to manipulate it algebraically.
We first convert into base e e as follows: 2^x = \left ( e^ { \ln 2 } \right) ^ x = e^ { x \ln 2 } . 2x = (eln2)x = exln2. Next, we apply the chain rule with f (x) = e^x f (x) = ex and g (x) = x \ln 2 g(x) = xln2 to obtain.
$$\frac{\text{d}}{\text{d}x}e^x=e^x$$ The "Chain" Rule. When the exponential expression is something other than simply x, we apply the chain rule: First we take the derivative of the entire expression, then we multiply it by the derivative of the expression in the exponent.
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