Calculus Examples

$y = a x^{4} + b x^{2} + c$

Step 1

Find the first derivative.

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Step 1.1

By the Sum Rule, the derivative of $a x^{4} + b x^{2} + c$ with respect to $a$ is $\frac{d}{d a} [a x^{4}] + \frac{d}{d a} [b x^{2}] + \frac{d}{d a} [c]$ .

$\frac{d}{d a} [a x^{4}] + \frac{d}{d a} [b x^{2}] + \frac{d}{d a} [c]$

Step 1.2

Evaluate $\frac{d}{d a} [a x^{4}]$ .

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Step 1.2.1

Since $x^{4}$ is constant with respect to $a$ , the derivative of $a x^{4}$ with respect to $a$ is $x^{4} \frac{d}{d a} [a]$ .

$x^{4} \frac{d}{d a} [a] + \frac{d}{d a} [b x^{2}] + \frac{d}{d a} [c]$

Step 1.2.2

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

$x^{4} \cdot 1 + \frac{d}{d a} [b x^{2}] + \frac{d}{d a} [c]$

Step 1.2.3

Multiply $x^{4}$ by $1$ .

$x^{4} + \frac{d}{d a} [b x^{2}] + \frac{d}{d a} [c]$

Step 1.3

Differentiate using the Constant Rule.

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Step 1.3.1

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

$x^{4} + 0 + \frac{d}{d a} [c]$

Step 1.3.2

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

$x^{4} + 0 + 0$

Step 1.4

Combine terms.

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Step 1.4.1

Add $x^{4}$ and $0$ .

$x^{4} + 0$

Step 1.4.2

Add $x^{4}$ and $0$ .

$f' (a) = x^{4}$

Step 2

Since $x^{4}$ is constant with respect to $a$ , the derivative of $x^{4}$ with respect to $a$ is $0$ .

$f'' (a) = 0$

Step 3

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

$f''' (a) = 0$