Calculus Examples

$\int_{0}^{3 x} \sqrt{1 + t^{2}} d t$

Step 1

Take the derivative of $\int_{0}^{3 x} \sqrt{1 + t^{2}} d t$ with respect to $x$ using Fundamental Theorem of Calculus and the chain rule.

$\frac{d}{d x} [3 x] \sqrt{1 + {(3 x)}^{2}}$

Step 2

Differentiate.

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Step 2.1

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

$3 \frac{d}{d x} [x] \sqrt{1 + {(3 x)}^{2}}$

Step 2.2

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

$3 \cdot 1 \sqrt{1 + {(3 x)}^{2}}$

Step 2.3

Simplify with factoring out.

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Step 2.3.1

Multiply $3$ by $1$ .

$3 \sqrt{1 + {(3 x)}^{2}}$

Step 2.3.2

Factor $3$ out of $3 x$ .

$3 \sqrt{1 + {(3 (x))}^{2}}$

Step 2.3.3

Simplify the expression.

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Step 2.3.3.1

Apply the product rule to $3 (x)$ .

$3 \sqrt{1 + 3^{2} x^{2}}$

Step 2.3.3.2

Raise $3$ to the power of $2$ .

$3 \sqrt{1 + 9 x^{2}}$