Calculus Examples

$P (θ (3)) - θ (P (3))$

Step 1

By the Sum Rule, the derivative of $P (θ (3)) - θ (P (3))$ with respect to $P$ is $\frac{d}{d P} [P (θ (3))] + \frac{d}{d P} [- θ (P (3))]$ .

$\frac{d}{d P} [P (θ (3))] + \frac{d}{d P} [- θ (P (3))]$

Step 2

Evaluate $\frac{d}{d P} [P (θ (3))]$ .

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Step 2.1

Move $3$ to the left of $θ$ .

$\frac{d}{d P} [P (3 θ)] + \frac{d}{d P} [- θ (P (3))]$

Step 2.2

Move $3$ to the left of $P$ .

$\frac{d}{d P} [3 \cdot P θ] + \frac{d}{d P} [- θ (P (3))]$

Step 2.3

Since $3 θ$ is constant with respect to $P$ , the derivative of $3 P θ$ with respect to $P$ is $3 θ \frac{d}{d P} [P]$ .

$3 θ \frac{d}{d P} [P] + \frac{d}{d P} [- θ (P (3))]$

Step 2.4

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

$3 θ \cdot 1 + \frac{d}{d P} [- θ (P (3))]$

Step 2.5

Multiply $3$ by $1$ .

$3 θ + \frac{d}{d P} [- θ (P (3))]$

Step 3

Evaluate $\frac{d}{d P} [- θ (P (3))]$ .

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Step 3.1

Move $3$ to the left of $P$ .

$3 θ + \frac{d}{d P} [- θ (3 P)]$

Step 3.2

Multiply $3$ by $- 1$ .

$3 θ + \frac{d}{d P} [- 3 θ P]$

Step 3.3

Since $- 3 θ$ is constant with respect to $P$ , the derivative of $- 3 θ P$ with respect to $P$ is $- 3 θ \frac{d}{d P} [P]$ .

$3 θ - 3 θ \frac{d}{d P} [P]$

Step 3.4

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

$3 θ - 3 θ \cdot 1$

Step 3.5

Multiply $- 3$ by $1$ .

$3 θ - 3 θ$

Step 4

Subtract $3 θ$ from $3 θ$ .

$0$