Calculus Examples

$f (x) = 10.72 {(0.09 t + 10)}^{0.3}$

Step 1

Consider the difference quotient formula.

$\frac{f (x + h) - f (x)}{h}$

Step 2

Find the components of the definition.

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Step 2.1

Evaluate the function at $x = x + h$ .

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Step 2.1.1

Replace the variable $x$ with $x + h$ in the expression.

$f (x + h) = 10.72 {(0.09 t + 10)}^{0.3}$

Step 2.1.2

The final answer is $10.72 {(0.09 t + 10)}^{0.3}$ .

$10.72 {(0.09 t + 10)}^{0.3}$

Step 2.2

Find the components of the definition.

$f (x + h) = 10.72 {(0.09 t + 10)}^{0.3}$

$f (x) = 10.72 {(0.09 t + 10)}^{0.3}$

$f (x + h) = 10.72 {(0.09 t + 10)}^{0.3}$

$f (x) = 10.72 {(0.09 t + 10)}^{0.3}$

Step 3

Plug in the components.

$\frac{f (x + h) - f (x)}{h} = \frac{10.72 {(0.09 t + 10)}^{0.3} - (10.72 {(0.09 t + 10)}^{0.3})}{h}$

Step 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

Factor $0.04 {(0.09 t + 10)}^{0.3}$ out of $10.72 {(0.09 t + 10)}^{0.3} - 1 \cdot 10.72 {(0.09 t + 10)}^{0.3}$ .

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Step 4.1.1.1

Factor $0.04 {(0.09 t + 10)}^{0.3}$ out of $10.72 {(0.09 t + 10)}^{0.3}$ .

$\frac{0.04 {(0.09 t + 10)}^{0.3} (268) - 1 \cdot 10.72 {(0.09 t + 10)}^{0.3}}{h}$

Step 4.1.1.2

Factor $0.04 {(0.09 t + 10)}^{0.3}$ out of $- 1 \cdot 10.72 {(0.09 t + 10)}^{0.3}$ .

$\frac{0.04 {(0.09 t + 10)}^{0.3} (268) + 0.04 {(0.09 t + 10)}^{0.3} (- 25 \cdot 10.72)}{h}$

Step 4.1.1.3

Factor $0.04 {(0.09 t + 10)}^{0.3}$ out of $0.04 {(0.09 t + 10)}^{0.3} (268) + 0.04 {(0.09 t + 10)}^{0.3} (- 25 \cdot 10.72)$ .

$\frac{0.04 {(0.09 t + 10)}^{0.3} (268 - 25 \cdot 10.72)}{h}$

Step 4.1.2

Combine exponents.

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Step 4.1.2.1

Multiply $- 25$ by $10.72$ .

$\frac{0.04 {(0.09 t + 10)}^{0.3} (268 - 268)}{h}$

Step 4.1.2.2

Multiply $268 - 268$ by $0.04$ .

$\frac{0 {(0.09 t + 10)}^{0.3}}{h}$

Step 4.1.2.3

Multiply $0$ by ${(0.09 t + 10)}^{0.3}$ .

$\frac{0}{h}$

Step 4.2

Divide $0$ by $h$ .

$0$

Step 5