Calculus Examples

$y = b \cos (t) + t^{2} \sin (t)$

Step 1

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

$\frac{d}{d b} [b \cos (t)] + \frac{d}{d b} [t^{2} \sin (t)]$

Step 2

Evaluate $\frac{d}{d b} [b \cos (t)]$ .

Tap for more steps...

Step 2.1

Since $\cos (t)$ is constant with respect to $b$ , the derivative of $b \cos (t)$ with respect to $b$ is $\cos (t) \frac{d}{d b} [b]$ .

$\cos (t) \frac{d}{d b} [b] + \frac{d}{d b} [t^{2} \sin (t)]$

Step 2.2

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

$\cos (t) \cdot 1 + \frac{d}{d b} [t^{2} \sin (t)]$

Step 2.3

Multiply $\cos (t)$ by $1$ .

$\cos (t) + \frac{d}{d b} [t^{2} \sin (t)]$

Step 3

Differentiate using the Constant Rule.

Tap for more steps...

Step 3.1

Since $t^{2} \sin (t)$ is constant with respect to $b$ , the derivative of $t^{2} \sin (t)$ with respect to $b$ is $0$ .

$\cos (t) + 0$

Step 3.2

Add $\cos (t)$ and $0$ .

$\cos (t)$