12.13 Derivative
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Let \(x \in \mathbb{R}\) and \(\alpha \in \mathbb{R}\), show that for the function \(f(x) = x^\alpha := e^{\alpha \ln(x)}\), its derivative function is \(f'(x) = \alpha x^{\alpha-1}\). Hint: Use Chain rule.
Since \(f(x) = x^\alpha = e^{\alpha \ln(x)}\), then \(f^{\prime}(x)=e^{\alpha\ln\left(x\right)}\cdot\alpha\cdot\frac{1}{x}=\alpha x^{\alpha-1}\)
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For \(a > 0\), prove that the derivative of the exponential function \(f(x) = a^x = e^{x \ln(a)}\) is \(f'(x) = e^{x \ln(a)} \ln(a)\). Determine the derivative of its inverse using the Inverse Function Theorem.
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Compute \(f'(x)\) in the following cases:
- \(f(x) = (x^3 - 2x + 1)^8\)
- \(f(x) = \cos(\sqrt{x^4 + 6})\)
- \(f(x) = \frac{\sqrt{1 + \sin(3x)}}{1 - x + x^5}\)
- \(f(x) = (x^2)^{x^3}\)
- \(f(x) = \sin^3(5x) \cos^2\left(\frac{x}{3}\right)\)