Chan Adkins
02/01/2023 · Primary School
Use the limit definition to find the derivative of the function. \[ f(t)=t^{3}+t^{2} \]
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To find the derivative of the function \( f(t) = t^{3} + t^{2} \) using the limit definition, we can use the formula:
\[ f'(t) = \lim_{h \to 0} \frac{f(t + h) - f(t)}{h} \]
Substitute the function \( f(t) = t^{3} + t^{2} \) into the formula:
\[ f'(t) = \lim_{h \to 0} \frac{(t + h)^{3} + (t + h)^{2} - (t^{3} + t^{2})}{h} \]
Now, we can simplify the expression inside the limit and calculate the derivative.
Evaluate the limit by following steps:
- step0: Evaluate using transformations:
\(\lim _{h\rightarrow 0}\left(\frac{\left(t+h\right)^{3}+\left(t+h\right)^{2}-\left(t^{3}+t^{2}\right)}{h}\right)\)
- step1: Calculate:
\(\lim _{h\rightarrow 0}\left(\frac{3ht^{2}+3h^{2}t+h^{3}+2ht+h^{2}}{h}\right)\)
- step2: Reduce the fraction:
\(\lim _{h\rightarrow 0}\left(3t^{2}+3ht+h^{2}+2t+h\right)\)
- step3: Rewrite the expression:
\(\lim _{h\rightarrow 0}\left(3t^{2}\right)+\lim _{h\rightarrow 0}\left(3ht\right)+\lim _{h\rightarrow 0}\left(h^{2}\right)+\lim _{h\rightarrow 0}\left(2t\right)+\lim _{h\rightarrow 0}\left(h\right)\)
- step4: Calculate:
\(3t^{2}+0+0+2t+0\)
- step5: Calculate:
\(3t^{2}+2t\)
The derivative of the function \( f(t) = t^{3} + t^{2} \) using the limit definition is \( f'(t) = 3t^{2} + 2t \).
Quick Answer
The derivative of \( f(t) = t^{3} + t^{2} \) is \( f'(t) = 3t^{2} + 2t \).
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