Find the Derivative - d/dt (50t^2)/(t^2+100)

$\frac{50 t^{2}}{t^{2} + 100}$

Step 1

Since $50$ is constant with respect to $t$ , the derivative of $\frac{50 t^{2}}{t^{2} + 100}$ with respect to $t$ is $50 \frac{d}{d t} [\frac{t^{2}}{t^{2} + 100}]$ .

$50 \frac{d}{d t} [\frac{t^{2}}{t^{2} + 100}]$

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 $g (t) = t^{2} + 100$ .

$50 \frac{(t^{2} + 100) \frac{d}{d t} [t^{2}] - t^{2} \frac{d}{d t} [t^{2} + 100]}{{(t^{2} + 100)}^{2}}$

Step 3

Differentiate.

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

$50 \frac{(t^{2} + 100) (2 t) - t^{2} \frac{d}{d t} [t^{2} + 100]}{{(t^{2} + 100)}^{2}}$

Move $2$ to the left of $t^{2} + 100$ .

$50 \frac{2 \cdot (t^{2} + 100) t - t^{2} \frac{d}{d t} [t^{2} + 100]}{{(t^{2} + 100)}^{2}}$

By the Sum Rule, the derivative of $t^{2} + 100$ with respect to $t$ is $\frac{d}{d t} [t^{2}] + \frac{d}{d t} [100]$ .

$50 \frac{2 (t^{2} + 100) t - t^{2} (\frac{d}{d t} [t^{2}] + \frac{d}{d t} [100])}{{(t^{2} + 100)}^{2}}$

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

$50 \frac{2 (t^{2} + 100) t - t^{2} (2 t + \frac{d}{d t} [100])}{{(t^{2} + 100)}^{2}}$

Since $100$ is constant with respect to $t$ , the derivative of $100$ with respect to $t$ is $0$ .

$50 \frac{2 (t^{2} + 100) t - t^{2} (2 t + 0)}{{(t^{2} + 100)}^{2}}$

Simplify the expression.

Add $2 t$ and $0$ .

$50 \frac{2 (t^{2} + 100) t - t^{2} (2 t)}{{(t^{2} + 100)}^{2}}$

Multiply $2$ by $- 1$ .

$50 \frac{2 (t^{2} + 100) t - 2 t^{2} t}{{(t^{2} + 100)}^{2}}$

Step 4

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

$50 \frac{2 (t^{2} + 100) t - 2 (t^{1} t^{2})}{{(t^{2} + 100)}^{2}}$

Step 5

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

$50 \frac{2 (t^{2} + 100) t - 2 t^{1 + 2}}{{(t^{2} + 100)}^{2}}$

Step 6

Add $1$ and $2$ .

$50 \frac{2 (t^{2} + 100) t - 2 t^{3}}{{(t^{2} + 100)}^{2}}$

Step 7

Combine $50$ and $\frac{2 (t^{2} + 100) t - 2 t^{3}}{{(t^{2} + 100)}^{2}}$ .

$\frac{50 (2 (t^{2} + 100) t - 2 t^{3})}{{(t^{2} + 100)}^{2}}$

Step 8

Simplify.

Apply the distributive property.

$\frac{50 ((2 t^{2} + 2 \cdot 100) t - 2 t^{3})}{{(t^{2} + 100)}^{2}}$

Apply the distributive property.

$\frac{50 (2 t^{2} t + 2 \cdot 100 t - 2 t^{3})}{{(t^{2} + 100)}^{2}}$

Apply the distributive property.

$\frac{50 (2 t^{2} t) + 50 (2 \cdot 100 t) + 50 (- 2 t^{3})}{{(t^{2} + 100)}^{2}}$

Simplify the numerator.

Simplify each term.

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

Move $t$ .

$\frac{50 (2 (t \cdot t^{2})) + 50 (2 \cdot 100 t) + 50 (- 2 t^{3})}{{(t^{2} + 100)}^{2}}$

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

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

$\frac{50 (2 (t^{1} t^{2})) + 50 (2 \cdot 100 t) + 50 (- 2 t^{3})}{{(t^{2} + 100)}^{2}}$

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

$\frac{50 (2 t^{1 + 2}) + 50 (2 \cdot 100 t) + 50 (- 2 t^{3})}{{(t^{2} + 100)}^{2}}$

Add $1$ and $2$ .

$\frac{50 (2 t^{3}) + 50 (2 \cdot 100 t) + 50 (- 2 t^{3})}{{(t^{2} + 100)}^{2}}$

Multiply $2$ by $50$ .

$\frac{100 t^{3} + 50 (2 \cdot 100 t) + 50 (- 2 t^{3})}{{(t^{2} + 100)}^{2}}$

Multiply $2$ by $100$ .

$\frac{100 t^{3} + 50 (200 t) + 50 (- 2 t^{3})}{{(t^{2} + 100)}^{2}}$

Multiply $200$ by $50$ .

$\frac{100 t^{3} + 10000 t + 50 (- 2 t^{3})}{{(t^{2} + 100)}^{2}}$

Multiply $- 2$ by $50$ .

$\frac{100 t^{3} + 10000 t - 100 t^{3}}{{(t^{2} + 100)}^{2}}$

Combine the opposite terms in $100 t^{3} + 10000 t - 100 t^{3}$ .

Subtract $100 t^{3}$ from $100 t^{3}$ .

$\frac{10000 t + 0}{{(t^{2} + 100)}^{2}}$

Add $10000 t$ and $0$ .

$\frac{10000 t}{{(t^{2} + 100)}^{2}}$

Find the Derivative - d/dt (50t^2)/(t^2+100)

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