Calculus Examples

$P (t) = (0.7 t - 6) (0.9 t + 9) + 95$

Step 1

By the Sum Rule, the derivative of $(0.7 t - 6) (0.9 t + 9) + 95$ with respect to $t$ is $\frac{d}{d t} [(0.7 t - 6) (0.9 t + 9)] + \frac{d}{d t} [95]$ .

$\frac{d}{d t} [(0.7 t - 6) (0.9 t + 9)] + \frac{d}{d t} [95]$

Step 2

Evaluate $\frac{d}{d t} [(0.7 t - 6) (0.9 t + 9)]$ .

Tap for more steps...

Step 2.1

Differentiate using the Product Rule which states that $\frac{d}{d t} [f (t) g (t)]$ is $f (t) \frac{d}{d t} [g (t)] + g (t) \frac{d}{d t} [f (t)]$ where $f (t) = 0.7 t - 6$ and $g (t) = 0.9 t + 9$ .

$(0.7 t - 6) \frac{d}{d t} [0.9 t + 9] + (0.9 t + 9) \frac{d}{d t} [0.7 t - 6] + \frac{d}{d t} [95]$

Step 2.2

By the Sum Rule, the derivative of $0.9 t + 9$ with respect to $t$ is $\frac{d}{d t} [0.9 t] + \frac{d}{d t} [9]$ .

$(0.7 t - 6) (\frac{d}{d t} [0.9 t] + \frac{d}{d t} [9]) + (0.9 t + 9) \frac{d}{d t} [0.7 t - 6] + \frac{d}{d t} [95]$

Step 2.3

Since $0.9$ is constant with respect to $t$ , the derivative of $0.9 t$ with respect to $t$ is $0.9 \frac{d}{d t} [t]$ .

$(0.7 t - 6) (0.9 \frac{d}{d t} [t] + \frac{d}{d t} [9]) + (0.9 t + 9) \frac{d}{d t} [0.7 t - 6] + \frac{d}{d t} [95]$

Step 2.4

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

$(0.7 t - 6) (0.9 \cdot 1 + \frac{d}{d t} [9]) + (0.9 t + 9) \frac{d}{d t} [0.7 t - 6] + \frac{d}{d t} [95]$

Step 2.5

Since $9$ is constant with respect to $t$ , the derivative of $9$ with respect to $t$ is $0$ .

$(0.7 t - 6) (0.9 \cdot 1 + 0) + (0.9 t + 9) \frac{d}{d t} [0.7 t - 6] + \frac{d}{d t} [95]$

Step 2.6

By the Sum Rule, the derivative of $0.7 t - 6$ with respect to $t$ is $\frac{d}{d t} [0.7 t] + \frac{d}{d t} [- 6]$ .

$(0.7 t - 6) (0.9 \cdot 1 + 0) + (0.9 t + 9) (\frac{d}{d t} [0.7 t] + \frac{d}{d t} [- 6]) + \frac{d}{d t} [95]$

Step 2.7

Since $0.7$ is constant with respect to $t$ , the derivative of $0.7 t$ with respect to $t$ is $0.7 \frac{d}{d t} [t]$ .

$(0.7 t - 6) (0.9 \cdot 1 + 0) + (0.9 t + 9) (0.7 \frac{d}{d t} [t] + \frac{d}{d t} [- 6]) + \frac{d}{d t} [95]$

Step 2.8

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

$(0.7 t - 6) (0.9 \cdot 1 + 0) + (0.9 t + 9) (0.7 \cdot 1 + \frac{d}{d t} [- 6]) + \frac{d}{d t} [95]$

Step 2.9

Since $- 6$ is constant with respect to $t$ , the derivative of $- 6$ with respect to $t$ is $0$ .

$(0.7 t - 6) (0.9 \cdot 1 + 0) + (0.9 t + 9) (0.7 \cdot 1 + 0) + \frac{d}{d t} [95]$

Step 2.10

Multiply $0.9$ by $1$ .

$(0.7 t - 6) (0.9 + 0) + (0.9 t + 9) (0.7 \cdot 1 + 0) + \frac{d}{d t} [95]$

Step 2.11

Add $0.9$ and $0$ .

$(0.7 t - 6) \cdot 0.9 + (0.9 t + 9) (0.7 \cdot 1 + 0) + \frac{d}{d t} [95]$

Step 2.12

Move $0.9$ to the left of $0.7 t - 6$ .

$0.9 \cdot (0.7 t - 6) + (0.9 t + 9) (0.7 \cdot 1 + 0) + \frac{d}{d t} [95]$

Step 2.13

Multiply $0.7$ by $1$ .

$0.9 (0.7 t - 6) + (0.9 t + 9) (0.7 + 0) + \frac{d}{d t} [95]$

Step 2.14

Add $0.7$ and $0$ .

$0.9 (0.7 t - 6) + (0.9 t + 9) \cdot 0.7 + \frac{d}{d t} [95]$

Step 2.15

Move $0.7$ to the left of $0.9 t + 9$ .

$0.9 (0.7 t - 6) + 0.7 (0.9 t + 9) + \frac{d}{d t} [95]$

Step 3

Since $95$ is constant with respect to $t$ , the derivative of $95$ with respect to $t$ is $0$ .

$0.9 (0.7 t - 6) + 0.7 (0.9 t + 9) + 0$

Step 4

Simplify.

Tap for more steps...

Step 4.1

Apply the distributive property.

$0.9 (0.7 t) + 0.9 \cdot - 6 + 0.7 (0.9 t + 9) + 0$

Step 4.2

Apply the distributive property.

$0.9 (0.7 t) + 0.9 \cdot - 6 + 0.7 (0.9 t) + 0.7 \cdot 9 + 0$

Step 4.3

Combine terms.

Tap for more steps...

Step 4.3.1

Multiply $0.7$ by $0.9$ .

$0.63 t + 0.9 \cdot - 6 + 0.7 (0.9 t) + 0.7 \cdot 9 + 0$

Step 4.3.2

Multiply $0.9$ by $- 6$ .

$0.63 t - 5.4 + 0.7 (0.9 t) + 0.7 \cdot 9 + 0$

Step 4.3.3

Multiply $0.9$ by $0.7$ .

$0.63 t - 5.4 + 0.63 t + 0.7 \cdot 9 + 0$

Step 4.3.4

Multiply $0.7$ by $9$ .

$0.63 t - 5.4 + 0.63 t + 6.3 + 0$

Step 4.3.5

Add $0.63 t$ and $0.63 t$ .

$1.26 t - 5.4 + 6.3 + 0$

Step 4.3.6

Add $- 5.4$ and $6.3$ .

$1.26 t + 0.9 + 0$

Step 4.3.7

Add $1.26 t + 0.9$ and $0$ .

$1.26 t + 0.9$