Calculus Mathematics Quiz: Differentiation Problems

1. Differentiate y=2x⁴-4x+5

8x4-4x

8x3-4

24x3+4

8x4+4x

None of the above

The given expression is y=2x^4-4x+5. To differentiate this expression, we need to find the derivative with respect to x. Taking the derivative of each term, we get 8x^3 for the first term, -4 for the second term (as the derivative of -4x is -4), and 0 for the constant term. Therefore, the correct answer is 8x^3-4, which matches the given answer.

Explanation

The given expression is y=2x^4-4x+5. To differentiate this expression, we need to find the derivative with respect to x. Taking the derivative of each term, we get 8x^3 for the first term, -4 for the second term (as the derivative of -4x is -4), and 0 for the constant term. Therefore, the correct answer is 8x^3-4, which matches the given answer.

2. Differentiate y= -6x⁴+3x²-12x

24x3-6x

-24x5+3x2-12x

-24x3+6x-12

64x3-6x

The given expression y = -6x^4 + 3x^2 - 12x can be differentiated using the power rule of differentiation. The power rule states that for a term of the form ax^n, the derivative is given by nax^(n-1). Applying this rule to each term in the expression, we get the derivative as -24x^3 + 6x - 12.

Explanation

The given expression y = -6x^4 + 3x^2 - 12x can be differentiated using the power rule of differentiation. The power rule states that for a term of the form ax^n, the derivative is given by nax^(n-1). Applying this rule to each term in the expression, we get the derivative as -24x^3 + 6x - 12.

3. Differentiate of x⁶ is 6x, true or false?

True

False

The statement is false because the given differentiation is incorrect. The correct differentiation of x^6 is 6x^5.

Explanation

The statement is false because the given differentiation is incorrect. The correct differentiation of x^6 is 6x^5.

4. Match the following

y=1-5x

y=5x-1

y=13x+2

y=6

y=3+6x

Submit

5. Differentiate and Match the following

y=cos(3x+1)

y=sin (1-5x)

y=2cos (6x+2)

y=sin (2x+7)

Submit

6. Differentiate of y= 3e^-3x is -9e^-3x true or false?

True

False

The given statement is true. The correct differentiation of y= 3e-3x is -9e-3x. This can be obtained by applying the power rule of differentiation, which states that the derivative of e^kx is k * e^kx. In this case, k is -3x, so the derivative becomes -3 * e^-3x. Simplifying further, we get -9e^-3x, which matches the given answer. Therefore, the statement is true.

Explanation

The given statement is true. The correct differentiation of y= 3e-3x is -9e-3x. This can be obtained by applying the power rule of differentiation, which states that the derivative of e^kx is k * e^kx. In this case, k is -3x, so the derivative becomes -3 * e^-3x. Simplifying further, we get -9e^-3x, which matches the given answer. Therefore, the statement is true.

7. Differentiate y= tan (3x²-1)

Sec2 (3x2-1)

6x sec2 (3x2-1)

3x sec2 (3x2-10

Sec2 (3x-1)

The correct answer is 6x sec^2(3x^2-1). To differentiate y = tan(3x^2-1), we can use the chain rule. The derivative of tan(u) is sec^2(u), and the derivative of (3x^2-1) with respect to x is 6x. Therefore, the derivative of y = tan(3x^2-1) is 6x sec^2(3x^2-1).

Explanation

The correct answer is 6x sec^2(3x^2-1). To differentiate y = tan(3x^2-1), we can use the chain rule. The derivative of tan(u) is sec^2(u), and the derivative of (3x^2-1) with respect to x is 6x. Therefore, the derivative of y = tan(3x^2-1) is 6x sec^2(3x^2-1).

8. Differentiate y=7x²- 3e^3x+1

14 - 9e3x+1

14x2-3e4x+1

14x - 3e

14x - 9e3x+1

None of the above

The given answer, 14x - 9e3x+1, is the correct differentiation of the function y=7x2 - 3e3x+1. The derivative of 7x2 is 14x, and the derivative of -3e3x+1 is -9e3x+1. Therefore, the correct answer is 14x - 9e3x+1.

Explanation

The given answer, 14x - 9e3x+1, is the correct differentiation of the function y=7x2 - 3e3x+1. The derivative of 7x2 is 14x, and the derivative of -3e3x+1 is -9e3x+1. Therefore, the correct answer is 14x - 9e3x+1.

9. Which of the following represents the derivative of f(x) = x^3 * e^x ?

3x^2 * e^x

X^3 * e^x + 3x^2 * e^x

X^3 * e^x - 3x^2 * e^x

E^x

Explanation

10. Differentiate y = x(3x²- 5)

9x2 -5x

9x - 5

6x

9x2 - 5

None of the above

The correct answer is 9x^2 - 5. This is the correct answer because when we differentiate the given function, we use the power rule of differentiation. The power rule states that when differentiating a function of the form x^n, the derivative is nx^(n-1). Applying this rule to the given function, we differentiate the term 3x^2 to get 6x, and the constant term -5 differentiates to 0. Therefore, the derivative of y = x(3x^2 - 5) is 3x^2 + 6x - 5. Simplifying this expression gives us the correct answer of 9x^2 - 5.

Explanation

The correct answer is 9x^2 - 5. This is the correct answer because when we differentiate the given function, we use the power rule of differentiation. The power rule states that when differentiating a function of the form x^n, the derivative is nx^(n-1). Applying this rule to the given function, we differentiate the term 3x^2 to get 6x, and the constant term -5 differentiates to 0. Therefore, the derivative of y = x(3x^2 - 5) is 3x^2 + 6x - 5. Simplifying this expression gives us the correct answer of 9x^2 - 5.