Calculus Examples

$f (x) = 4 (0^{\frac{5}{4}}) + 4 (0^{\frac{3}{2}}) + 10 x$

Step 1

Differentiate using the Sum Rule.

Tap for more steps...

Step 1.1

Rewrite $0$ as $0^{4}$ .

$\frac{d}{d x} [4 {(0^{4})}^{\frac{5}{4}} + 4 (0^{\frac{3}{2}}) + 10 x]$

Step 1.2

Reduce the expression by cancelling the common factors.

Tap for more steps...

Step 1.2.1

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

$\frac{d}{d x} [4 \cdot 0^{4 (\frac{5}{4})} + 4 (0^{\frac{3}{2}}) + 10 x]$

Step 1.2.2

Cancel the common factor of $4$ .

Tap for more steps...

Step 1.2.2.1

Cancel the common factor.

$\frac{d}{d x} [4 \cdot 0^{4 (\frac{5}{4})} + 4 (0^{\frac{3}{2}}) + 10 x]$

Step 1.2.2.2

Rewrite the expression.

$\frac{d}{d x} [4 \cdot 0^{5} + 4 (0^{\frac{3}{2}}) + 10 x]$

Step 1.2.3

Simplify the expression.

Tap for more steps...

Step 1.2.3.1

Raising $0$ to any positive power yields $0$ .

$\frac{d}{d x} [4 \cdot 0 + 4 (0^{\frac{3}{2}}) + 10 x]$

Step 1.2.3.2

Rewrite $0$ as $0^{2}$ .

$\frac{d}{d x} [4 \cdot 0 + 4 {(0^{2})}^{\frac{3}{2}} + 10 x]$

Step 1.2.3.3

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

$\frac{d}{d x} [4 \cdot 0 + 4 \cdot 0^{2 (\frac{3}{2})} + 10 x]$

Step 1.2.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 1.2.4.1

Cancel the common factor.

$\frac{d}{d x} [4 \cdot 0 + 4 \cdot 0^{2 (\frac{3}{2})} + 10 x]$

Step 1.2.4.2

Rewrite the expression.

$\frac{d}{d x} [4 \cdot 0 + 4 \cdot 0^{3} + 10 x]$

Step 1.2.5

Raising $0$ to any positive power yields $0$ .

$\frac{d}{d x} [4 \cdot 0 + 4 \cdot 0 + 10 x]$

Step 1.3

By the Sum Rule, the derivative of $4 \cdot 0 + 4 \cdot 0 + 10 x$ with respect to $x$ is $\frac{d}{d x} [4 \cdot 0] + \frac{d}{d x} [4 \cdot 0] + \frac{d}{d x} [10 x]$ .

$\frac{d}{d x} [4 \cdot 0] + \frac{d}{d x} [4 \cdot 0] + \frac{d}{d x} [10 x]$

Step 2

Evaluate $\frac{d}{d x} [4 \cdot 0]$ .

Tap for more steps...

Step 2.1

Multiply $4$ by $0$ .

$\frac{d}{d x} [0] + \frac{d}{d x} [4 \cdot 0] + \frac{d}{d x} [10 x]$

Step 2.2

Since $0$ is constant with respect to $x$ , the derivative of $0$ with respect to $x$ is $0$ .

$0 + \frac{d}{d x} [4 \cdot 0] + \frac{d}{d x} [10 x]$

Step 3

Evaluate $\frac{d}{d x} [4 \cdot 0]$ .

Tap for more steps...

Step 3.1

Multiply $4$ by $0$ .

$0 + \frac{d}{d x} [0] + \frac{d}{d x} [10 x]$

Step 3.2

Since $0$ is constant with respect to $x$ , the derivative of $0$ with respect to $x$ is $0$ .

$0 + 0 + \frac{d}{d x} [10 x]$

Step 4

Evaluate $\frac{d}{d x} [10 x]$ .

Tap for more steps...

Step 4.1

Since $10$ is constant with respect to $x$ , the derivative of $10 x$ with respect to $x$ is $10 \frac{d}{d x} [x]$ .

$0 + 0 + 10 \frac{d}{d x} [x]$

Step 4.2

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

$0 + 0 + 10 \cdot 1$

Step 4.3

Multiply $10$ by $1$ .

$0 + 0 + 10$

Step 5

Combine terms.

Tap for more steps...

Step 5.1

Add $0$ and $0$ .

$0 + 10$

Step 5.2

Add $0$ and $10$ .

$10$