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By computing the first few derivatives and looking for a pattern, find the following derivatives.
c. d⁷³/dx⁷³ (x sin x)
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3. Techniques of Differentiation
Higher Order Derivatives
Scelta multipla
Find the third derivative of the given function.
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Start by identifying the function you need to differentiate: \( f(t) = 5t^{-4} \). This function is in the form of a power function, which is suitable for applying the power rule for differentiation.
Apply the power rule for differentiation to find the first derivative. The power rule states that if \( f(t) = at^n \), then \( f'(t) = a \cdot n \cdot t^{n-1} \). For \( f(t) = 5t^{-4} \), the first derivative \( f'(t) \) will be \( 5 \cdot (-4) \cdot t^{-5} \).
Next, find the second derivative by differentiating \( f'(t) = -20t^{-5} \) using the power rule again. The second derivative \( f''(t) \) will be \( -20 \cdot (-5) \cdot t^{-6} \).
Proceed to find the third derivative by differentiating \( f''(t) = 100t^{-6} \). Apply the power rule once more: \( f'''(t) = 100 \cdot (-6) \cdot t^{-7} \).
Simplify the expression for the third derivative: \( f'''(t) = -600t^{-7} \). This is the third derivative of the function \( f(t) = 5t^{-4} \).
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