Question

Find the derivative of y = 4x^3.

Find the derivative of y = 2/x^4.
Find the derivative of y = 3√x.
Find the derivative of y = 3/√x.
Find the derivative of y = x^3/4.
Find the derivative of y = 3 x^5.​

Answer 1

To find the derivatives of the given functions, we can use the power rule. Applying the power rule, we obtain the derivatives as follows: 1. 12x^2, 2. -8/x^5, 3. 3/(2√x), 4. 3/(2x√x), 5. (3/4)*x^(3/4), 6. 15x^4.

Step-by-step explanation:

To find the derivative of a function, we can use the power rule. The power rule states that the derivative of x^n is n*x^(n-1). Let's apply this rule to each of the given functions:

1. y = 4x^3. Using the power rule, the derivative of this function is 12x^2.

2. y = 2/x^4. Using the power rule, the derivative of this function is -8/x^5.

3. y = 3√x. Using the power rule, the derivative of this function is (1/2)*(3/x^(1/2)) = 3/(2√x).

4. y = 3/√x. Using the power rule, the derivative of this function is (1/2)*(3/x^(3/2)) = 3/(2x√x).

5. y = x^3/4. Using the power rule, the derivative of this function is (3/4)*x^(3/4).

6. y = 3x^5. Using the power rule, the derivative of this function is 15x^4.

Answer 2

`\fbox {See below} \downarrow`

Step-by-step explanation:

Question 1 :

`\mathsf {(dy)/(dx) = (d)/(dx)(4x^(3))}`

`\mathsf {(dy)/(dx) = 4(3)x^(3-1)}`

`\mathsf {(dy)/(dx) = 12x^(2)}`

Question 2 :

`\mathsf {(dy)/(dx) = (d)/(dx)((2)/(x^(4)))}`

`\mathsf {(dy)/(dx) = (d)/(dx)(2x^(-4))}`

`\mathsf {(dy)/(dx) = 2(-4)x^(-4-1)}`

`\mathsf {(dy)/(dx) = -8x^(-5)}`

`\mathsf {(dy)/(dx) = -(8)/(x^(5))}`

Question 3 :

`\mathsf {(dy)/(dx) = (d)/(dx)(3√(x))}`

`\mathsf {(dy)/(dx) = (d)/(dx)(3(x)^{(1)/(2)})}`

`\mathsf {(dy)/(dx) = 3((1)/(2))x^{(1)/(2)-1}}`

`\mathsf {(dy)/(dx) = (3)/(2)x^{-(1)/(2)}}`

`\mathsf {(dy)/(dx) = (3)/(2√(x))}`

Question 4 :

`\mathsf {(dy)/(dx) = (d)/(dx)((3)/(√(x)))}`

`\mathsf {(dy)/(dx) = (d)/(dx)(3x^{-(1)/(2)})}`

`\mathsf {(dy)/(dx) = 3(-(1)/(2))x^{-(1)/(2)-1}}`

`\mathsf {(dy)/(dx) = -(3)/(2)x^{-(3)/(2)}}`

`\mathsf {(dy)/(dx) = -(3)/(2x√(x))}`

Question 5 :

`\mathsf {(dy)/(dx) = (d)/(dx)((x^(3))/(4))}`

`\mathsf {(dy)/(dx) = (1)/(4)(3)x^(3-1)}`

`\mathsf {(dy)/(dx) = (3)/(4)x^(2)}`

Question 6 :

`\mathsf {(dy)/(dx) = (d)/(dx)(3x^(5))}`

`\mathsf {(dy)/(dx) = 3(5)x^(5-1)}`

`\mathsf {(dy)/(dx) = 15x^(4)}`