Calculus Examples

$f (y) = 4 x^{2} - 2 y^{3} + 24 x + 24 y - 35$

Step 1

Differentiate.

Tap for more steps...

Step 1.1

By the Sum Rule, the derivative of $4 x^{2} - 2 y^{3} + 24 x + 24 y - 35$ with respect to $y$ is $\frac{d}{d y} [4 x^{2}] + \frac{d}{d y} [- 2 y^{3}] + \frac{d}{d y} [24 x] + \frac{d}{d y} [24 y] + \frac{d}{d y} [- 35]$ .

$\frac{d}{d y} [4 x^{2}] + \frac{d}{d y} [- 2 y^{3}] + \frac{d}{d y} [24 x] + \frac{d}{d y} [24 y] + \frac{d}{d y} [- 35]$

Step 1.2

Since $4 x^{2}$ is constant with respect to $y$ , the derivative of $4 x^{2}$ with respect to $y$ is $0$ .

$0 + \frac{d}{d y} [- 2 y^{3}] + \frac{d}{d y} [24 x] + \frac{d}{d y} [24 y] + \frac{d}{d y} [- 35]$

Step 2

Evaluate $\frac{d}{d y} [- 2 y^{3}]$ .

Tap for more steps...

Step 2.1

Since $- 2$ is constant with respect to $y$ , the derivative of $- 2 y^{3}$ with respect to $y$ is $- 2 \frac{d}{d y} [y^{3}]$ .

$0 - 2 \frac{d}{d y} [y^{3}] + \frac{d}{d y} [24 x] + \frac{d}{d y} [24 y] + \frac{d}{d y} [- 35]$

Step 2.2

Differentiate using the Power Rule which states that $\frac{d}{d y} [y^{n}]$ is $n y^{n - 1}$ where $n = 3$ .

$0 - 2 (3 y^{2}) + \frac{d}{d y} [24 x] + \frac{d}{d y} [24 y] + \frac{d}{d y} [- 35]$

Step 2.3

Multiply $3$ by $- 2$ .

$0 - 6 y^{2} + \frac{d}{d y} [24 x] + \frac{d}{d y} [24 y] + \frac{d}{d y} [- 35]$

Step 3

Since $24 x$ is constant with respect to $y$ , the derivative of $24 x$ with respect to $y$ is $0$ .

$0 - 6 y^{2} + 0 + \frac{d}{d y} [24 y] + \frac{d}{d y} [- 35]$

Step 4

Evaluate $\frac{d}{d y} [24 y]$ .

Tap for more steps...

Step 4.1

Since $24$ is constant with respect to $y$ , the derivative of $24 y$ with respect to $y$ is $24 \frac{d}{d y} [y]$ .

$0 - 6 y^{2} + 0 + 24 \frac{d}{d y} [y] + \frac{d}{d y} [- 35]$

Step 4.2

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

$0 - 6 y^{2} + 0 + 24 \cdot 1 + \frac{d}{d y} [- 35]$

Step 4.3

Multiply $24$ by $1$ .

$0 - 6 y^{2} + 0 + 24 + \frac{d}{d y} [- 35]$

Step 5

Since $- 35$ is constant with respect to $y$ , the derivative of $- 35$ with respect to $y$ is $0$ .

$0 - 6 y^{2} + 0 + 24 + 0$

Step 6

Combine terms.

Tap for more steps...

Step 6.1

Subtract $6 y^{2}$ from $0$ .

$- 6 y^{2} + 0 + 24 + 0$

Step 6.2

Add $- 6 y^{2}$ and $0$ .

$- 6 y^{2} + 24 + 0$

Step 6.3

Add $- 6 y^{2} + 24$ and $0$ .

$- 6 y^{2} + 24$