Calculus Examples

$P' (t) = \frac{d}{d t} {(0.97)}^{t}$

Step 1

Find the first derivative.

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Step 1.1

Simplify terms.

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Step 1.1.1

Cancel the common factor of $d$ .

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Step 1.1.1.1

Cancel the common factor.

$\frac{d}{d t} [\frac{d}{d t} \cdot {(0.97)}^{t}]$

Step 1.1.1.2

Rewrite the expression.

$\frac{d}{d t} [\frac{1}{t} \cdot {(0.97)}^{t}]$

Step 1.1.2

Combine $\frac{1}{t}$ and ${(0.97)}^{t}$ .

$\frac{d}{d t} [\frac{{(0.97)}^{t}}{t}]$

Step 1.2

Differentiate using the Quotient Rule which states that $\frac{d}{d t} [\frac{f (t)}{g (t)}]$ is $\frac{g (t) \frac{d}{d t} [f (t)] - f (t) \frac{d}{d t} [g (t)]}{{g (t)}^{2}}$ where $f (t) = {0.97}^{t}$ and $g (t) = t$ .

$\frac{t \frac{d}{d t} [{0.97}^{t}] - {0.97}^{t} \frac{d}{d t} [t]}{t^{2}}$

Step 1.3

Differentiate using the Exponential Rule which states that $\frac{d}{d t} [a^{t}]$ is $a^{t} \ln (a)$ where $a$ = $0.97$ .

$\frac{t ({0.97}^{t} \ln (0.97)) - {0.97}^{t} \frac{d}{d t} [t]}{t^{2}}$

Step 1.4

Differentiate using the Power Rule.

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Step 1.4.1

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

$\frac{t ({0.97}^{t} \ln (0.97)) - {0.97}^{t} \cdot 1}{t^{2}}$

Step 1.4.2

Simplify the expression.

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Step 1.4.2.1

Multiply $- 1$ by $1$ .

$\frac{t ({0.97}^{t} \ln (0.97)) - 1 \cdot {0.97}^{t}}{t^{2}}$

Step 1.4.2.2

Rewrite $- 1 \cdot {0.97}^{t}$ as $- {0.97}^{t}$ .

$\frac{t ({0.97}^{t} \ln (0.97)) - {0.97}^{t}}{t^{2}}$

Step 1.5

Simplify.

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Step 1.5.1

Simplify the numerator.

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Step 1.5.1.1

Rewrite using the commutative property of multiplication.

$\frac{\ln (0.97) t \cdot {0.97}^{t} - {0.97}^{t}}{t^{2}}$

Step 1.5.1.2

Reorder factors in $\ln (0.97) t \cdot {0.97}^{t} - {0.97}^{t}$ .

$\frac{t \cdot {0.97}^{t} \ln (0.97) - {0.97}^{t}}{t^{2}}$

Step 1.5.2

Reorder terms.

$\frac{{0.97}^{t} t \ln (0.97) - {0.97}^{t}}{t^{2}}$

Step 1.5.3

Factor ${0.97}^{t}$ out of ${0.97}^{t} t \ln (0.97) - {0.97}^{t}$ .

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Step 1.5.3.1

Factor ${0.97}^{t}$ out of ${0.97}^{t} t \ln (0.97)$ .

$\frac{{0.97}^{t} (t \ln (0.97)) - {0.97}^{t}}{t^{2}}$

Step 1.5.3.2

Factor ${0.97}^{t}$ out of $- {0.97}^{t}$ .

$\frac{{0.97}^{t} (t \ln (0.97)) + {0.97}^{t} \cdot - 1}{t^{2}}$

Step 1.5.3.3

Factor ${0.97}^{t}$ out of ${0.97}^{t} (t \ln (0.97)) + {0.97}^{t} \cdot - 1$ .

$f' (t) = \frac{{0.97}^{t} (t \ln (0.97) - 1)}{t^{2}}$

Step 2

Find the second derivative.

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Step 2.1

$\frac{t^{2} \frac{d}{d t} [{0.97}^{t} (t \ln (0.97) - 1)] - {0.97}^{t} (t \ln (0.97) - 1) \frac{d}{d t} [t^{2}]}{{(t^{2})}^{2}}$

Step 2.2

Multiply the exponents in ${(t^{2})}^{2}$ .

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Step 2.2.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{t^{2} \frac{d}{d t} [{0.97}^{t} (t \ln (0.97) - 1)] - {0.97}^{t} (t \ln (0.97) - 1) \frac{d}{d t} [t^{2}]}{t^{2 \cdot 2}}$

Step 2.2.2

Multiply $2$ by $2$ .

$\frac{t^{2} \frac{d}{d t} [{0.97}^{t} (t \ln (0.97) - 1)] - {0.97}^{t} (t \ln (0.97) - 1) \frac{d}{d t} [t^{2}]}{t^{4}}$

Step 2.3

Differentiate using the Product Rule which states that $\frac{d}{d t} [f (t) g (t)]$ is $f (t) \frac{d}{d t} [g (t)] + g (t) \frac{d}{d t} [f (t)]$ where $f (t) = {0.97}^{t}$ and $g (t) = t \ln (0.97) - 1$ .

$\frac{t^{2} ({0.97}^{t} \frac{d}{d t} [t \ln (0.97) - 1] + (t \ln (0.97) - 1) \frac{d}{d t} [{0.97}^{t}]) - {0.97}^{t} (t \ln (0.97) - 1) \frac{d}{d t} [t^{2}]}{t^{4}}$

Step 2.4

Differentiate.

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Step 2.4.1

By the Sum Rule, the derivative of $t \ln (0.97) - 1$ with respect to $t$ is $\frac{d}{d t} [t \ln (0.97)] + \frac{d}{d t} [- 1]$ .

$\frac{t^{2} ({0.97}^{t} (\frac{d}{d t} [t \ln (0.97)] + \frac{d}{d t} [- 1]) + (t \ln (0.97) - 1) \frac{d}{d t} [{0.97}^{t}]) - {0.97}^{t} (t \ln (0.97) - 1) \frac{d}{d t} [t^{2}]}{t^{4}}$

Step 2.4.2

Since $\ln (0.97)$ is constant with respect to $t$ , the derivative of $t \ln (0.97)$ with respect to $t$ is $\ln (0.97) \frac{d}{d t} [t]$ .

$\frac{t^{2} ({0.97}^{t} (\ln (0.97) \frac{d}{d t} [t] + \frac{d}{d t} [- 1]) + (t \ln (0.97) - 1) \frac{d}{d t} [{0.97}^{t}]) - {0.97}^{t} (t \ln (0.97) - 1) \frac{d}{d t} [t^{2}]}{t^{4}}$

Step 2.4.3

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

$\frac{t^{2} ({0.97}^{t} (\ln (0.97) \cdot 1 + \frac{d}{d t} [- 1]) + (t \ln (0.97) - 1) \frac{d}{d t} [{0.97}^{t}]) - {0.97}^{t} (t \ln (0.97) - 1) \frac{d}{d t} [t^{2}]}{t^{4}}$

Step 2.4.4

Multiply $\ln (0.97)$ by $1$ .

$\frac{t^{2} ({0.97}^{t} (\ln (0.97) + \frac{d}{d t} [- 1]) + (t \ln (0.97) - 1) \frac{d}{d t} [{0.97}^{t}]) - {0.97}^{t} (t \ln (0.97) - 1) \frac{d}{d t} [t^{2}]}{t^{4}}$

Step 2.4.5

Since $- 1$ is constant with respect to $t$ , the derivative of $- 1$ with respect to $t$ is $0$ .

$\frac{t^{2} ({0.97}^{t} (\ln (0.97) + 0) + (t \ln (0.97) - 1) \frac{d}{d t} [{0.97}^{t}]) - {0.97}^{t} (t \ln (0.97) - 1) \frac{d}{d t} [t^{2}]}{t^{4}}$

Step 2.4.6

Add $\ln (0.97)$ and $0$ .

$\frac{t^{2} ({0.97}^{t} \ln (0.97) + (t \ln (0.97) - 1) \frac{d}{d t} [{0.97}^{t}]) - {0.97}^{t} (t \ln (0.97) - 1) \frac{d}{d t} [t^{2}]}{t^{4}}$

Step 2.5

Differentiate using the Exponential Rule which states that $\frac{d}{d t} [a^{t}]$ is $a^{t} \ln (a)$ where $a$ = $0.97$ .

$\frac{t^{2} ({0.97}^{t} \ln (0.97) + (t \ln (0.97) - 1) ({0.97}^{t} \ln (0.97))) - {0.97}^{t} (t \ln (0.97) - 1) \frac{d}{d t} [t^{2}]}{t^{4}}$

Step 2.6

Differentiate using the Power Rule.

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Step 2.6.1

Differentiate using the Power Rule which states that $\frac{d}{d t} [t^{n}]$ is $n t^{n - 1}$ where $n = 2$ .

$\frac{t^{2} ({0.97}^{t} \ln (0.97) + (t \ln (0.97) - 1) \cdot {0.97}^{t} \ln (0.97)) - {0.97}^{t} (t \ln (0.97) - 1) (2 t)}{t^{4}}$

Step 2.6.2

Simplify with factoring out.

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Step 2.6.2.1

Multiply $2$ by $- 1$ .

$\frac{t^{2} ({0.97}^{t} \ln (0.97) + (t \ln (0.97) - 1) \cdot {0.97}^{t} \ln (0.97)) - 2 \cdot {0.97}^{t} (t \ln (0.97) - 1) t}{t^{4}}$

Step 2.6.2.2

Factor $t$ out of $t^{2} ({0.97}^{t} \ln (0.97) + (t \ln (0.97) - 1) \cdot {0.97}^{t} \ln (0.97)) - 2 \cdot {0.97}^{t} (t \ln (0.97) - 1) t$ .

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Step 2.6.2.2.1

Factor $t$ out of $t^{2} ({0.97}^{t} \ln (0.97) + (t \ln (0.97) - 1) \cdot {0.97}^{t} \ln (0.97))$ .

$\frac{t (t ({0.97}^{t} \ln (0.97) + (t \ln (0.97) - 1) \cdot {0.97}^{t} \ln (0.97))) - 2 \cdot {0.97}^{t} (t \ln (0.97) - 1) t}{t^{4}}$

Step 2.6.2.2.2

Factor $t$ out of $- 2 \cdot {0.97}^{t} (t \ln (0.97) - 1) t$ .

$\frac{t (t ({0.97}^{t} \ln (0.97) + (t \ln (0.97) - 1) \cdot {0.97}^{t} \ln (0.97))) + t (- 2 \cdot {0.97}^{t} (t \ln (0.97) - 1))}{t^{4}}$

Step 2.6.2.2.3

Factor $t$ out of $t (t ({0.97}^{t} \ln (0.97) + (t \ln (0.97) - 1) \cdot {0.97}^{t} \ln (0.97))) + t (- 2 \cdot {0.97}^{t} (t \ln (0.97) - 1))$ .

$\frac{t (t ({0.97}^{t} \ln (0.97) + (t \ln (0.97) - 1) \cdot {0.97}^{t} \ln (0.97)) - 2 \cdot {0.97}^{t} (t \ln (0.97) - 1))}{t^{4}}$

Step 2.7

Cancel the common factors.

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Step 2.7.1

Factor $t$ out of $t^{4}$ .

$\frac{t (t ({0.97}^{t} \ln (0.97) + (t \ln (0.97) - 1) \cdot {0.97}^{t} \ln (0.97)) - 2 \cdot {0.97}^{t} (t \ln (0.97) - 1))}{t \cdot t^{3}}$

Step 2.7.2

Cancel the common factor.

$\frac{t (t ({0.97}^{t} \ln (0.97) + (t \ln (0.97) - 1) \cdot {0.97}^{t} \ln (0.97)) - 2 \cdot {0.97}^{t} (t \ln (0.97) - 1))}{t \cdot t^{3}}$

Step 2.7.3

Rewrite the expression.

$\frac{t ({0.97}^{t} \ln (0.97) + (t \ln (0.97) - 1) \cdot {0.97}^{t} \ln (0.97)) - 2 \cdot {0.97}^{t} (t \ln (0.97) - 1)}{t^{3}}$

Step 2.8

Simplify.

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Step 2.8.1

Apply the distributive property.

$\frac{t ({0.97}^{t} \ln (0.97) + (t \ln (0.97) \cdot {0.97}^{t} - 1 \cdot {0.97}^{t}) \ln (0.97)) - 2 \cdot {0.97}^{t} (t \ln (0.97) - 1)}{t^{3}}$

Step 2.8.2

Apply the distributive property.

$\frac{t ({0.97}^{t} \ln (0.97) + t \ln (0.97) \cdot {0.97}^{t} \ln (0.97) - 1 \cdot {0.97}^{t} \ln (0.97)) - 2 \cdot {0.97}^{t} (t \ln (0.97) - 1)}{t^{3}}$

Step 2.8.3

Apply the distributive property.

$\frac{t ({0.97}^{t} \ln (0.97)) + t (t \ln (0.97) \cdot {0.97}^{t} \ln (0.97)) + t (- 1 \cdot {0.97}^{t} \ln (0.97)) - 2 \cdot {0.97}^{t} (t \ln (0.97) - 1)}{t^{3}}$

Step 2.8.4

Apply the distributive property.

$\frac{t ({0.97}^{t} \ln (0.97)) + t (t \ln (0.97) \cdot {0.97}^{t} \ln (0.97)) + t (- 1 \cdot {0.97}^{t} \ln (0.97)) - 2 \cdot {0.97}^{t} (t \ln (0.97)) - 2 \cdot {0.97}^{t} \cdot - 1}{t^{3}}$

Step 2.8.5

Simplify the numerator.

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Step 2.8.5.1

Combine the opposite terms in $t ({0.97}^{t} \ln (0.97)) + t (t \ln (0.97) \cdot {0.97}^{t} \ln (0.97)) + t (- 1 \cdot {0.97}^{t} \ln (0.97)) - 2 \cdot {0.97}^{t} (t \ln (0.97)) - 2 \cdot {0.97}^{t} \cdot - 1$ .

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Step 2.8.5.1.1

Reorder the factors in the terms $t ({0.97}^{t} \ln (0.97))$ and $t (- 1 \cdot {0.97}^{t} \ln (0.97))$ .

$\frac{{0.97}^{t} t \ln (0.97) + t (t \ln (0.97) \cdot {0.97}^{t} \ln (0.97)) - {0.97}^{t} t \ln (0.97) - 2 \cdot {0.97}^{t} (t \ln (0.97)) - 2 \cdot {0.97}^{t} \cdot - 1}{t^{3}}$

Step 2.8.5.1.2

Subtract ${0.97}^{t} t \ln (0.97)$ from ${0.97}^{t} t \ln (0.97)$ .

$\frac{t (t \ln (0.97) \cdot {0.97}^{t} \ln (0.97)) + 0 - 2 \cdot {0.97}^{t} (t \ln (0.97)) - 2 \cdot {0.97}^{t} \cdot - 1}{t^{3}}$

Step 2.8.5.1.3

Add $t (t \ln (0.97) \cdot {0.97}^{t} \ln (0.97))$ and $0$ .

$\frac{t (t \ln (0.97) \cdot {0.97}^{t} \ln (0.97)) - 2 \cdot {0.97}^{t} (t \ln (0.97)) - 2 \cdot {0.97}^{t} \cdot - 1}{t^{3}}$

Step 2.8.5.2

Simplify each term.

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Step 2.8.5.2.1

Rewrite using the commutative property of multiplication.

$\frac{\ln (0.97) \ln (0.97) t (t \cdot {0.97}^{t}) - 2 \cdot {0.97}^{t} (t \ln (0.97)) - 2 \cdot {0.97}^{t} \cdot - 1}{t^{3}}$

Step 2.8.5.2.2

Multiply $t$ by $t$ by adding the exponents.

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Step 2.8.5.2.2.1

Move $t$ .

$\frac{\ln (0.97) \ln (0.97) (t \cdot t) \cdot {0.97}^{t} - 2 \cdot {0.97}^{t} (t \ln (0.97)) - 2 \cdot {0.97}^{t} \cdot - 1}{t^{3}}$

Step 2.8.5.2.2.2

Multiply $t$ by $t$ .

$\frac{\ln (0.97) \ln (0.97) t^{2} \cdot {0.97}^{t} - 2 \cdot {0.97}^{t} (t \ln (0.97)) - 2 \cdot {0.97}^{t} \cdot - 1}{t^{3}}$

Step 2.8.5.2.3

Multiply $\ln (0.97) \ln (0.97)$ .

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Step 2.8.5.2.3.1

Raise $\ln (0.97)$ to the power of $1$ .

$\frac{\ln^{1} (0.97) \ln (0.97) t^{2} \cdot {0.97}^{t} - 2 \cdot {0.97}^{t} (t \ln (0.97)) - 2 \cdot {0.97}^{t} \cdot - 1}{t^{3}}$

Step 2.8.5.2.3.2

Raise $\ln (0.97)$ to the power of $1$ .

$\frac{\ln^{1} (0.97) \ln^{1} (0.97) t^{2} \cdot {0.97}^{t} - 2 \cdot {0.97}^{t} (t \ln (0.97)) - 2 \cdot {0.97}^{t} \cdot - 1}{t^{3}}$

Step 2.8.5.2.3.3

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

$\frac{{\ln (0.97)}^{1 + 1} t^{2} \cdot {0.97}^{t} - 2 \cdot {0.97}^{t} (t \ln (0.97)) - 2 \cdot {0.97}^{t} \cdot - 1}{t^{3}}$

Step 2.8.5.2.3.4

Add $1$ and $1$ .

$\frac{\ln^{2} (0.97) t^{2} \cdot {0.97}^{t} - 2 \cdot {0.97}^{t} (t \ln (0.97)) - 2 \cdot {0.97}^{t} \cdot - 1}{t^{3}}$

Step 2.8.5.2.4

Simplify $- 2 \ln (0.97)$ by moving $2$ inside the logarithm.

$\frac{\ln^{2} (0.97) t^{2} \cdot {0.97}^{t} + {0.97}^{t} (t (- \ln ({0.97}^{2}))) - 2 \cdot {0.97}^{t} \cdot - 1}{t^{3}}$

Step 2.8.5.2.5

Rewrite using the commutative property of multiplication.

$\frac{\ln^{2} (0.97) t^{2} \cdot {0.97}^{t} - \ln ({0.97}^{2}) \cdot {0.97}^{t} t - 2 \cdot {0.97}^{t} \cdot - 1}{t^{3}}$

Step 2.8.5.2.6

Raise $0.97$ to the power of $2$ .

$\frac{\ln^{2} (0.97) t^{2} \cdot {0.97}^{t} - \ln (0.9409) \cdot {0.97}^{t} t - 2 \cdot {0.97}^{t} \cdot - 1}{t^{3}}$

Step 2.8.5.2.7

Multiply $- 1$ by $- 2$ .

$\frac{\ln^{2} (0.97) t^{2} \cdot {0.97}^{t} - \ln (0.9409) \cdot {0.97}^{t} t + 2 \cdot {0.97}^{t}}{t^{3}}$

Step 2.8.5.3

Reorder factors in $\ln^{2} (0.97) t^{2} \cdot {0.97}^{t} - \ln (0.9409) \cdot {0.97}^{t} t + 2 \cdot {0.97}^{t}$ .

$\frac{t^{2} \cdot {0.97}^{t} \ln^{2} (0.97) - t \cdot {0.97}^{t} \ln (0.9409) + 2 \cdot {0.97}^{t}}{t^{3}}$

Step 2.8.6

Reorder terms.

$\frac{{0.97}^{t} t^{2} \ln^{2} (0.97) - {0.97}^{t} t \ln (0.9409) + 2 \cdot {0.97}^{t}}{t^{3}}$

Step 2.8.7

Reorder factors in $\frac{{0.97}^{t} t^{2} \ln^{2} (0.97) - {0.97}^{t} t \ln (0.9409) + 2 \cdot {0.97}^{t}}{t^{3}}$ .

$f'' (t) = \frac{t^{2} \cdot {0.97}^{t} \ln^{2} (0.97) - t \cdot {0.97}^{t} \ln (0.9409) + 2 \cdot {0.97}^{t}}{t^{3}}$

Step 3

The second derivative of $P' (t)$ with respect to $t$ is $\frac{t^{2} \cdot ({0.97}^{t} \ln^{2} (0.97)) - t \cdot ({0.97}^{t} \ln (0.9409)) + 2 \cdot {0.97}^{t}}{t^{3}}$ .

$\frac{t^{2} \cdot ({0.97}^{t} \ln^{2} (0.97)) - t \cdot ({0.97}^{t} \ln (0.9409)) + 2 \cdot {0.97}^{t}}{t^{3}}$