Find the Derivative - d/dd d/(dt)*6(t^2+5t)^-7

$\frac{d}{d t} \cdot 6 {(t^{2} + 5 t)}^{- 7}$

Step 1

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

$\frac{d}{d \cdot d} [\frac{d}{d t} \cdot (6 \frac{1}{{(t^{2} + 5 t)}^{7}})]$

Step 2

Simplify the denominator.

Factor $t$ out of $t^{2} + 5 t$ .

Factor $t$ out of $t^{2}$ .

$\frac{d}{d \cdot d} [\frac{d}{d t} \cdot (6 \frac{1}{{(t \cdot t + 5 t)}^{7}})]$

Factor $t$ out of $5 t$ .

$\frac{d}{d \cdot d} [\frac{d}{d t} \cdot (6 \frac{1}{{(t \cdot t + t \cdot 5)}^{7}})]$

Factor $t$ out of $t \cdot t + t \cdot 5$ .

$\frac{d}{d \cdot d} [\frac{d}{d t} \cdot (6 \frac{1}{{(t (t + 5))}^{7}})]$

Apply the product rule to $t (t + 5)$ .

$\frac{d}{d \cdot d} [\frac{d}{d t} \cdot (6 \frac{1}{t^{7} {(t + 5)}^{7}})]$

Step 3

Simplify terms.

Cancel the common factor of $d$ .

Cancel the common factor.

$\frac{d}{d \cdot d} [\frac{d}{d t} \cdot (6 \frac{1}{t^{7} {(t + 5)}^{7}})]$

Rewrite the expression.

$\frac{d}{d \cdot d} [\frac{1}{t} \cdot (6 \frac{1}{t^{7} {(t + 5)}^{7}})]$

Combine $6$ and $\frac{1}{t^{7} {(t + 5)}^{7}}$ .

$\frac{d}{d \cdot d} [\frac{1}{t} \cdot \frac{6}{t^{7} {(t + 5)}^{7}}]$

Multiply $\frac{1}{t}$ by $\frac{6}{t^{7} {(t + 5)}^{7}}$ .

$\frac{d}{d \cdot d} [\frac{6}{t (t^{7} {(t + 5)}^{7})}]$

Step 4

Raise $t$ to the power of $1$ .

$\frac{d}{d \cdot d} [\frac{6}{t^{7} t^{1} {(t + 5)}^{7}}]$

Step 5

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{d}{d \cdot d} [\frac{6}{t^{7 + 1} {(t + 5)}^{7}}]$

Step 6

Add $7$ and $1$ .

$\frac{d}{d \cdot d} [\frac{6}{t^{8} {(t + 5)}^{7}}]$

Step 7

Since $\frac{6}{t^{8} {(t + 5)}^{7}}$ is constant with respect to $d$ , the derivative of $\frac{6}{t^{8} {(t + 5)}^{7}}$ with respect to $d$ is $0$ .

$0$

Find the Derivative - d/dd d/(dt)*6(t^2+5t)^-7

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