Math / Calculus

QuestionAlso given is $y=-8 t^{2}+t^{3}$ .
Differentiate $y$ with respect to $t$ . $\frac{d y}{d t}=$ $\square$

Studdy Solution

STEP 1

1. We are given the function $y = -8t^2 + t^3$ .
2. We need to differentiate $y$ with respect to $t$ .

STEP 2

1. Identify each term in the function $y$ .
2. Apply the power rule of differentiation to each term.
3. Combine the results to find the derivative $\frac{dy}{dt}$ .

STEP 3

Identify each term in the function $y$ :
The function is $y = -8t^2 + t^3$ .

STEP 4

Apply the power rule to the first term $-8t^2$ :
The power rule states that $\frac{d}{dt}(t^n) = nt^{n-1}$ .
For $-8t^2$ , differentiate as follows:
$\frac{d}{dt}(-8t^2) = -8 \cdot 2t^{2-1} = -16t$

STEP 5

Apply the power rule to the second term $t^3$ :
For $t^3$ , differentiate as follows:
$\frac{d}{dt}(t^3) = 3t^{3-1} = 3t^2$

STEP 6

Combine the results from the differentiation of each term:
$\frac{dy}{dt} = -16t + 3t^2$
The derivative of $y$ with respect to $t$ is:
$\boxed{-16t + 3t^2}$

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QuestionAlso given is $y=-8 t^{2}+t^{3}$ .
Differentiate $y$ with respect to $t$ . $\frac{d y}{d t}=$ $\square$

Studdy Solution

STEP 1

1. We are given the function $y = -8t^2 + t^3$ .
2. We need to differentiate $y$ with respect to $t$ .

STEP 2

1. Identify each term in the function $y$ .
2. Apply the power rule of differentiation to each term.
3. Combine the results to find the derivative $\frac{dy}{dt}$ .

STEP 3

Identify each term in the function $y$ :
The function is $y = -8t^2 + t^3$ .

STEP 4

Apply the power rule to the first term $-8t^2$ :
The power rule states that $\frac{d}{dt}(t^n) = nt^{n-1}$ .
For $-8t^2$ , differentiate as follows:
$\frac{d}{dt}(-8t^2) = -8 \cdot 2t^{2-1} = -16t$

STEP 5

Apply the power rule to the second term $t^3$ :
For $t^3$ , differentiate as follows:
$\frac{d}{dt}(t^3) = 3t^{3-1} = 3t^2$

STEP 6

Combine the results from the differentiation of each term:
$\frac{dy}{dt} = -16t + 3t^2$
The derivative of $y$ with respect to $t$ is:
$\boxed{-16t + 3t^2}$

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QuestionAlso given is y=−8t2+t3y=-8 t^{2}+t^{3}y=−8t2+t3.Differentiate yyy with respect to ttt. dydt=\frac{d y}{d t}=dtdy​= □\square□

Studdy Solution

STEP 1

1. We are given the function y=−8t2+t3 y = -8t^2 + t^3 y=−8t2+t3.2. We need to differentiate y y y with respect to t t t.

STEP 2

1. Identify each term in the function y y y.2. Apply the power rule of differentiation to each term.3. Combine the results to find the derivative dydt\frac{dy}{dt}dtdy​.

STEP 3

Identify each term in the function y y y:The function is y=−8t2+t3 y = -8t^2 + t^3 y=−8t2+t3.

STEP 4

STEP 5

Apply the power rule to the second term t3t^3t3:For t3t^3t3, differentiate as follows:ddt(t3)=3t3−1=3t2 \frac{d}{dt}(t^3) = 3t^{3-1} = 3t^2 dtd​(t3)=3t3−1=3t2

STEP 6

Combine the results from the differentiation of each term:dydt=−16t+3t2 \frac{dy}{dt} = -16t + 3t^2 dtdy​=−16t+3t2 The derivative of y y y with respect to t t t is:−16t+3t2 \boxed{-16t + 3t^2} −16t+3t2​

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QuestionAlso given is $y=-8 t^{2}+t^{3}$ .
Differentiate $y$ with respect to $t$ . $\frac{d y}{d t}=$ $\square$

1. We are given the function $y = -8t^2 + t^3$ .
2. We need to differentiate $y$ with respect to $t$ .

1. Identify each term in the function $y$ .
2. Apply the power rule of differentiation to each term.
3. Combine the results to find the derivative $\frac{dy}{dt}$ .

Identify each term in the function $y$ :
The function is $y = -8t^2 + t^3$ .

Apply the power rule to the second term $t^3$ :
For $t^3$ , differentiate as follows:
$\frac{d}{dt}(t^3) = 3t^{3-1} = 3t^2$

Combine the results from the differentiation of each term:
$\frac{dy}{dt} = -16t + 3t^2$
The derivative of $y$ with respect to $t$ is:
$\boxed{-16t + 3t^2}$