Calculus Examples

$\frac{d}{(d x) \int_{0}^{5 x^{2}} \frac{t}{t + 21} d t}$

Step 1

Cancel the common factor of $d$ .

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Step 1.1

Cancel the common factor.

$\frac{d}{d x} [\frac{d}{d x \int_{0}^{5 x^{2}} \frac{t}{t + 21} d t}]$

Step 1.2

Rewrite the expression.

$\frac{d}{d x} [\frac{1}{x \int_{0}^{5 x^{2}} \frac{t}{t + 21} d t}]$

Step 2

Since $\frac{1}{\int_{0}^{5 x^{2}} \frac{t}{t + 21} d t}$ is constant with respect to $x$ , the derivative of $\frac{1}{x \int_{0}^{5 x^{2}} \frac{t}{t + 21} d t}$ with respect to $x$ is $\frac{1}{\int_{0}^{5 x^{2}} \frac{t}{t + 21} d t} \frac{d}{d x} [\frac{1}{x}]$ .

$\frac{1}{\int_{0}^{5 x^{2}} \frac{t}{t + 21} d t} \frac{d}{d x} [\frac{1}{x}]$

Step 3

Rewrite $\frac{1}{x}$ as $x^{- 1}$ .

$\frac{1}{\int_{0}^{5 x^{2}} \frac{t}{t + 21} d t} \frac{d}{d x} [x^{- 1}]$

Step 4

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

$\frac{1}{\int_{0}^{5 x^{2}} \frac{t}{t + 21} d t} (- x^{- 2})$

Step 5

Simplify.

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Step 5.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{1}{\int_{0}^{5 x^{2}} \frac{t}{t + 21} d t} (- \frac{1}{x^{2}})$

Step 5.2

Reorder the factors of $\frac{1}{\int_{0}^{5 x^{2}} \frac{t}{t + 21} d t} (- \frac{1}{x^{2}})$ .

$- \frac{1}{x^{2}} \cdot \frac{1}{\int_{0}^{5 x^{2}} \frac{t}{t + 21} d t}$

Step 5.3

Combine $- \frac{1}{x^{2}}$ and $\frac{1}{\int_{0}^{5 x^{2}} \frac{t}{t + 21} d t}$ .

$- \frac{1}{x^{2} \int_{0}^{5 x^{2}} \frac{t}{t + 21} d t}$