Calculus Examples

$\frac{d}{d x} \int_{7}^{x^{3}} \frac{1}{p^{2}} d p$

Step 1

Cancel the common factor of $d$ .

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Cancel the common factor.

$\frac{d}{d x} [\frac{d}{d x} \cdot \int_{7}^{x^{3}} \frac{1}{p^{2}} d p]$

Rewrite the expression.

$\frac{d}{d x} [\frac{1}{x} \cdot \int_{7}^{x^{3}} \frac{1}{p^{2}} d p]$

Step 2

Since $\int_{7}^{x^{3}} \frac{1}{p^{2}} d p$ is constant with respect to $x$ , the derivative of $\frac{1}{x} \int_{7}^{x^{3}} \frac{1}{p^{2}} d p$ with respect to $x$ is $\int_{7}^{x^{3}} \frac{1}{p^{2}} d p \frac{d}{d x} [\frac{1}{x}]$ .

$\int_{7}^{x^{3}} \frac{1}{p^{2}} d p \frac{d}{d x} [\frac{1}{x}]$

Step 3

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

$\int_{7}^{x^{3}} \frac{1}{p^{2}} d p \frac{d}{d x} [x^{- 1}]$

Step 4

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

$\int_{7}^{x^{3}} \frac{1}{p^{2}} d p (- x^{- 2})$

Step 5

Simplify.

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

$\int_{7}^{x^{3}} \frac{1}{p^{2}} d p (- \frac{1}{x^{2}})$

Reorder the factors of $\int_{7}^{x^{3}} \frac{1}{p^{2}} d p (- \frac{1}{x^{2}})$ .

$- \frac{1}{x^{2}} \int_{7}^{x^{3}} \frac{1}{p^{2}} d p$

Combine $- \frac{1}{x^{2}}$ and $\int_{7}^{x^{3}} \frac{1}{p^{2}} d p$ .

$- \frac{\int_{7}^{x^{3}} \frac{1}{p^{2}} d p}{x^{2}}$