Calculus Examples

$f (x) = a x^{3} + b x^{2} + c x + d$

Step 1

By the Sum Rule, the derivative of $a x^{3} + b x^{2} + c x + d$ with respect to $a$ is $\frac{d}{d a} [a x^{3}] + \frac{d}{d a} [b x^{2}] + \frac{d}{d a} [c x] + \frac{d}{d a} [d]$ .

$\frac{d}{d a} [a x^{3}] + \frac{d}{d a} [b x^{2}] + \frac{d}{d a} [c x] + \frac{d}{d a} [d]$

Step 2

Evaluate $\frac{d}{d a} [a x^{3}]$ .

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Step 2.1

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

$x^{3} \frac{d}{d a} [a] + \frac{d}{d a} [b x^{2}] + \frac{d}{d a} [c x] + \frac{d}{d a} [d]$

Step 2.2

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

$x^{3} \cdot 1 + \frac{d}{d a} [b x^{2}] + \frac{d}{d a} [c x] + \frac{d}{d a} [d]$

Step 2.3

Multiply $x^{3}$ by $1$ .

$x^{3} + \frac{d}{d a} [b x^{2}] + \frac{d}{d a} [c x] + \frac{d}{d a} [d]$

Step 3

Differentiate using the Constant Rule.

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Step 3.1

Since $b x^{2}$ is constant with respect to $a$ , the derivative of $b x^{2}$ with respect to $a$ is $0$ .

$x^{3} + 0 + \frac{d}{d a} [c x] + \frac{d}{d a} [d]$

Step 3.2

Since $c x$ is constant with respect to $a$ , the derivative of $c x$ with respect to $a$ is $0$ .

$x^{3} + 0 + 0 + \frac{d}{d a} [d]$

Step 3.3

Since $d$ is constant with respect to $a$ , the derivative of $d$ with respect to $a$ is $0$ .

$x^{3} + 0 + 0 + 0$

Step 4

Combine terms.

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Step 4.1

Add $x^{3}$ and $0$ .

$x^{3} + 0 + 0$

Step 4.2

Add $x^{3}$ and $0$ .

$x^{3} + 0$

Step 4.3

Add $x^{3}$ and $0$ .

$x^{3}$