Find the Derivative - d/dt (60t^2)/(t^2+150)

$\frac{60 t^{2}}{t^{2} + 150}$

Step 1

Since $60$ is constant with respect to $t$ , the derivative of $\frac{60 t^{2}}{t^{2} + 150}$ with respect to $t$ is $60 \frac{d}{d t} [\frac{t^{2}}{t^{2} + 150}]$ .

$60 \frac{d}{d t} [\frac{t^{2}}{t^{2} + 150}]$

Step 2

Differentiate using the Quotient Rule which states that $\frac{d}{d t} [\frac{f (t)}{g (t)}]$ is $\frac{g (t) \frac{d}{d t} [f (t)] - f (t) \frac{d}{d t} [g (t)]}{{g (t)}^{2}}$ where $f (t) = t^{2}$ and $g (t) = t^{2} + 150$ .

$60 \frac{(t^{2} + 150) \frac{d}{d t} [t^{2}] - t^{2} \frac{d}{d t} [t^{2} + 150]}{{(t^{2} + 150)}^{2}}$

Step 3

Differentiate.

Differentiate using the Power Rule which states that $\frac{d}{d t} [t^{n}]$ is $n t^{n - 1}$ where $n = 2$ .

$60 \frac{(t^{2} + 150) (2 t) - t^{2} \frac{d}{d t} [t^{2} + 150]}{{(t^{2} + 150)}^{2}}$

Move $2$ to the left of $t^{2} + 150$ .

$60 \frac{2 \cdot (t^{2} + 150) t - t^{2} \frac{d}{d t} [t^{2} + 150]}{{(t^{2} + 150)}^{2}}$

By the Sum Rule, the derivative of $t^{2} + 150$ with respect to $t$ is $\frac{d}{d t} [t^{2}] + \frac{d}{d t} [150]$ .

$60 \frac{2 (t^{2} + 150) t - t^{2} (\frac{d}{d t} [t^{2}] + \frac{d}{d t} [150])}{{(t^{2} + 150)}^{2}}$

Differentiate using the Power Rule which states that $\frac{d}{d t} [t^{n}]$ is $n t^{n - 1}$ where $n = 2$ .

$60 \frac{2 (t^{2} + 150) t - t^{2} (2 t + \frac{d}{d t} [150])}{{(t^{2} + 150)}^{2}}$

Since $150$ is constant with respect to $t$ , the derivative of $150$ with respect to $t$ is $0$ .

$60 \frac{2 (t^{2} + 150) t - t^{2} (2 t + 0)}{{(t^{2} + 150)}^{2}}$

Simplify the expression.

Add $2 t$ and $0$ .

$60 \frac{2 (t^{2} + 150) t - t^{2} (2 t)}{{(t^{2} + 150)}^{2}}$

Multiply $2$ by $- 1$ .

$60 \frac{2 (t^{2} + 150) t - 2 t^{2} t}{{(t^{2} + 150)}^{2}}$

Step 4

Raise $t$ to the power of $1$ .

$60 \frac{2 (t^{2} + 150) t - 2 (t^{1} t^{2})}{{(t^{2} + 150)}^{2}}$

Step 5

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$60 \frac{2 (t^{2} + 150) t - 2 t^{1 + 2}}{{(t^{2} + 150)}^{2}}$

Step 6

Add $1$ and $2$ .

$60 \frac{2 (t^{2} + 150) t - 2 t^{3}}{{(t^{2} + 150)}^{2}}$

Step 7

Combine $60$ and $\frac{2 (t^{2} + 150) t - 2 t^{3}}{{(t^{2} + 150)}^{2}}$ .

$\frac{60 (2 (t^{2} + 150) t - 2 t^{3})}{{(t^{2} + 150)}^{2}}$

Step 8

Simplify.

Apply the distributive property.

$\frac{60 ((2 t^{2} + 2 \cdot 150) t - 2 t^{3})}{{(t^{2} + 150)}^{2}}$

Apply the distributive property.

$\frac{60 (2 t^{2} t + 2 \cdot 150 t - 2 t^{3})}{{(t^{2} + 150)}^{2}}$

Apply the distributive property.

$\frac{60 (2 t^{2} t) + 60 (2 \cdot 150 t) + 60 (- 2 t^{3})}{{(t^{2} + 150)}^{2}}$

Simplify the numerator.

Simplify each term.

Multiply $t^{2}$ by $t$ by adding the exponents.

Move $t$ .

$\frac{60 (2 (t \cdot t^{2})) + 60 (2 \cdot 150 t) + 60 (- 2 t^{3})}{{(t^{2} + 150)}^{2}}$

Multiply $t$ by $t^{2}$ .

Raise $t$ to the power of $1$ .

$\frac{60 (2 (t^{1} t^{2})) + 60 (2 \cdot 150 t) + 60 (- 2 t^{3})}{{(t^{2} + 150)}^{2}}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{60 (2 t^{1 + 2}) + 60 (2 \cdot 150 t) + 60 (- 2 t^{3})}{{(t^{2} + 150)}^{2}}$

Add $1$ and $2$ .

$\frac{60 (2 t^{3}) + 60 (2 \cdot 150 t) + 60 (- 2 t^{3})}{{(t^{2} + 150)}^{2}}$

Multiply $2$ by $60$ .

$\frac{120 t^{3} + 60 (2 \cdot 150 t) + 60 (- 2 t^{3})}{{(t^{2} + 150)}^{2}}$

Multiply $2$ by $150$ .

$\frac{120 t^{3} + 60 (300 t) + 60 (- 2 t^{3})}{{(t^{2} + 150)}^{2}}$

Multiply $300$ by $60$ .

$\frac{120 t^{3} + 18000 t + 60 (- 2 t^{3})}{{(t^{2} + 150)}^{2}}$

Multiply $- 2$ by $60$ .

$\frac{120 t^{3} + 18000 t - 120 t^{3}}{{(t^{2} + 150)}^{2}}$

Combine the opposite terms in $120 t^{3} + 18000 t - 120 t^{3}$ .

Subtract $120 t^{3}$ from $120 t^{3}$ .

$\frac{18000 t + 0}{{(t^{2} + 150)}^{2}}$

Add $18000 t$ and $0$ .

$\frac{18000 t}{{(t^{2} + 150)}^{2}}$

Find the Derivative - d/dt (60t^2)/(t^2+150)

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