Calculus Examples

$y = (A x^{2} + B x + C) e^{5 x}$

Step 1

Find the first derivative.

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Step 1.1

Since $e^{5 x}$ is constant with respect to $A$ , the derivative of $(A x^{2} + B x + C) e^{5 x}$ with respect to $A$ is $e^{5 x} \frac{d}{d A} [A x^{2} + B x + C]$ .

$e^{5 x} \frac{d}{d A} [A x^{2} + B x + C]$

Step 1.2

By the Sum Rule, the derivative of $A x^{2} + B x + C$ with respect to $A$ is $\frac{d}{d A} [A x^{2}] + \frac{d}{d A} [B x] + \frac{d}{d A} [C]$ .

$e^{5 x} (\frac{d}{d A} [A x^{2}] + \frac{d}{d A} [B x] + \frac{d}{d A} [C])$

Step 1.3

Since $x^{2}$ is constant with respect to $A$ , the derivative of $A x^{2}$ with respect to $A$ is $x^{2} \frac{d}{d A} [A]$ .

$e^{5 x} (x^{2} \frac{d}{d A} [A] + \frac{d}{d A} [B x] + \frac{d}{d A} [C])$

Step 1.4

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

$e^{5 x} (x^{2} \cdot 1 + \frac{d}{d A} [B x] + \frac{d}{d A} [C])$

Step 1.5

Multiply $x^{2}$ by $1$ .

$e^{5 x} (x^{2} + \frac{d}{d A} [B x] + \frac{d}{d A} [C])$

Step 1.6

Since $B x$ is constant with respect to $A$ , the derivative of $B x$ with respect to $A$ is $0$ .

$e^{5 x} (x^{2} + 0 + \frac{d}{d A} [C])$

Step 1.7

Add $x^{2}$ and $0$ .

$e^{5 x} (x^{2} + \frac{d}{d A} [C])$

Step 1.8

Since $C$ is constant with respect to $A$ , the derivative of $C$ with respect to $A$ is $0$ .

$e^{5 x} (x^{2} + 0)$

Step 1.9

Simplify the expression.

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Step 1.9.1

Add $x^{2}$ and $0$ .

$e^{5 x} x^{2}$

Step 1.9.2

Reorder factors in $e^{5 x} x^{2}$ .

$f' (A) = x^{2} e^{5 x}$

Step 2

Since $x^{2} e^{5 x}$ is constant with respect to $A$ , the derivative of $x^{2} e^{5 x}$ with respect to $A$ is $0$ .

$f'' (A) = 0$