Question:

Find the derivative of the following: (5x^3+3x−1)(x−1).

Answer:

(5x^3+3x−1)(x−1)

= (5x^3+3x−1)(1) + (5x^3+3x−1)(-1)

= (5x^3+3x−1) + (-5x^3-3x+1)

= 0 + 0

= 0

Derivative of (5x^3+3x−1)(x−1) = 0

Question:

Find the derivative of the following: x^(−4)(3−4x^(−5)).

Answer:

Step 1: Use the power rule to rewrite the expression as: x^(-4 + (-5))(3 - 4x^(-5))

Step 2: Apply the power rule to the first term: (-4 + (-5))x^(-9)

Step 3: Apply the power rule to the second term: -4(-5)x^(-6)

Step 4: Combine the results: (-9)x^(-9) - 4(-5)x^(-6)

Step 5: Simplify: -9x^(-9) + 20x^(-6)

Question:

Find the derivative of the following functions:2tanx−7secx

Answer:

1. Use the Chain Rule: d/dx[2tanx - 7secx] = (2)(sec^2x) - (7)(secxtanx)

2. Simplify: (2)(sec^2x) - (7)(secxtanx) = 2sec^2x - 7sec^2x = -5sec^2x

Question:

Find the derivative of x at x=1

Answer:

Step 1: Begin with the equation for the derivative of a function f(x): f’(x) = lim (h→0) (f(x+h) - f(x))/h

Step 2: Substitute x=1 into the equation for the derivative of a function f(x): f’(1) = lim (h→0) (f(1+h) - f(1))/h

Step 3: Evaluate the limit as h approaches 0: f’(1) = lim (h→0) ((1+h) - 1)/h

Step 4: Simplify the equation: f’(1) = lim (h→0) (h/h)

Step 5: Evaluate the limit as h approaches 0: f’(1) = lim (h→0) 1

Step 6: The answer is: f’(1) = 1

Question:

For some constant a and b, find the derivative of the following functions: (ax^2+b)^2.

Answer:

Answer: Step 1: Use the power rule to find the derivative of (ax^2+b)^2.

Derivative = 2(ax^2+b)^2 * (2ax)

Step 2: Simplify the expression.

Derivative = 4a(ax^2+b)^2x

Question:

Find the derivatives of the following: secx.

Answer:

1. Use the chain rule: dy/dx = dy/du * du/dx

2. Identify the function: y = secx

3. Identify the derivative of the inner function: du/dx = tanx

4. Substitute into the chain rule: dy/dx = secx * tanx

5. Simplify: dy/dx = secxtanx

Question:

Find the derivative of the following functions: sin x cos x

Answer:

1. Use the product rule: d/dx (sin x cos x) = (d/dx sin x) cos x + (d/dx cos x) sin x

2. Use the chain rule: d/dx (sin x cos x) = (cos x)(d/dx sin x) + (sin x)(d/dx cos x)

3. Use the derivatives of sin x and cos x: d/dx (sin x cos x) = (cos x)(cos x) + (sin x)(-sin x)

4. Simplify: d/dx (sin x cos x) = cos2x - sin2x

Question:

Find the derivative of x^(−3)(5+3x)

Answer:

Answer: Step 1: Take the natural log of both sides: ln(x^(-3)(5+3x))

Step 2: Use the product rule to take the derivative: (-3x^(-4))(5+3x) + x^(-3)(3)

Step 3: Simplify: -3x^(-4)(5+3x) + 3x^(-3)

Step 4: Simplify further: -15x^(-4) + 3x^(-3)

Question:

Find the derivative of the following functions: 5secx+4cosx

Answer:

Answer: Step 1: Take the derivative of 5secx Derivative of 5secx = 5secx tanx

Step 2: Take the derivative of 4cosx Derivative of 4cosx = -4sinx

Step 3: Combine the derivatives Derivative of 5secx + 4cosx = 5secx tanx - 4sinx

Question:

Find the derivative of the following functions from first principle x^3−27

Answer:

Answer: Step 1: Identify the given function as f(x) = x^3 − 27

Step 2: Take the derivative of the function using the first principle, i.e., using the power rule

d/dx (f(x)) = d/dx (x^3 − 27)

Step 3: Apply the power rule:

d/dx (x^3 − 27) = 3x^2 (derivative of x^3) − 0 (derivative of 27)

Step 4: Simplify the expression:

d/dx (x^3 − 27) = 3x^2

Therefore, the derivative of the given function f(x) = x^3 − 27 is 3x^2.

Question:

For any constant real number a, find the derivative of: x^n+ax^(n−1)+a^2x^(n−2)+…+a^(n−1)x+a^n.

Answer:

Answer: Step 1: Use the power rule of derivatives to find the derivative of each term:

d/dx[x^n] = nx^(n-1)

d/dx[ax^(n-1)] = (n-1)ax^(n-2)

d/dx[a^2x^(n-2)] = (n-2)a^2x^(n-3)

d/dx[a^(n-1)x] = (n-1)a^(n-1)

d/dx[a^n] = 0

Step 2: Combine the derivatives to get the derivative of the entire expression:

d/dx[x^n+ax^(n−1)+a^2x^(n−2)+…+a^(n−1)x+a^n] = nx^(n-1) + (n-1)ax^(n-2) + (n-2)a^2x^(n-3) + … + (n-1)a^(n-1)

Question:

Find the derivative of the following functions: cosecx

Answer:

Answer: Step 1: Use the chain rule to rewrite the function as: cosecx = 1/sinx

Step 2: Take the derivative of 1/sinx using the quotient rule: d/dx (1/sinx) = -cosx/sinx^2

Step 3: Rewrite the expression using the cosecant function: d/dx (cosecx) = -cosecx * cotx

Question:

Find the derivative of x^2 - 2 at x = 10

Answer:

1. Identify the function: x^2 - 2

2. Take the derivative of the function: 2x

3. Substitute x = 10 into the derivative: 2(10) = 20

Question:

If f(x)=x^100/100+x^99/99+x^98/98 +…+x+1,showthatf’(1)=100 f’(0)$$.

Answer:

f’(x) = 100x^99/100 + 99x^98/99 + 98x^97/98 + … + 1

f’(1) = 100x^99/100 + 99x^98/99 + 98x^97/98 + … + 1

f’(1) = 100

f’(0) = 0x^99/100 + 99x^98/99 + 98x^97/98 + … + 1

f’(0) = 1

Question:

For some constant a and b, find the derivative of the following functions: (x−a)(x−b).

Answer:

Answer: Step 1: Rewrite the function as: (x-a)(x-b) = x^2 - (a+b)x + ab

Step 2: Take the derivative of the function: 2x - (a+b)

Step 3: Simplify the derivative: 2x - (a+b) = 2x - a - b

Question:

Find the derivatives of the following: 5sinx−6cosx+7.

Answer:

Answer:

1. Use the Chain Rule: d/dx[5sinx−6cosx+7] = d/dx[5sinx] - d/dx[6cosx] + d/dx[7]

2. Use the derivatives of sine and cosine: d/dx[5sinx] = 5cosx d/dx[6cosx] = -6sinx d/dx[7] = 0

3. Substitute the derivatives into the original equation: d/dx[5sinx−6cosx+7] = 5cosx - 6sinx + 0

4. Simplify: d/dx[5sinx−6cosx+7] = 5cosx - 6sinx

Question:

Find the derivative of 2x−3/4

Answer:

Answer:

1. Rewrite the expression in terms of a fraction: (2x - 3)/4
2. Take the derivative of numerator and denominator separately: Numerator: 2 Denominator: 0
3. Combine the two derivatives: 2/0
4. Simplify: Undefined

Question:

For some constant a and b, find the derivative of the following functions: (x−a)/(x−b).

Answer:

Answer: Step 1: Rewrite the function as: (x - a)(x - b)^-1

Step 2: Use the power rule to take the derivative of the function: -1(x - a)(x - b)^-2 + (x - b)^-1

Step 3: Simplify the derivative: (b - a)(x - b)^-2

Question:

Find the derivative of the following: x^5(3−6x^(−9)).

Answer:

Step 1: Rewrite the expression as x^14 - 6x^5.

Step 2: Use the power rule to find the derivative.

Step 3: The derivative of x^14 is 14x^13.

Step 4: The derivative of -6x^5 is -30x^4.

Step 5: Combine the two derivatives to get the final answer: 14x^13 - 30x^4.

Question:

Find the derivative of

Answer:

y = x^2

1. Rewrite the equation as y’ = (x^2)'

2. Apply the power rule: (x^2)’ = 2x^(2-1)

3. Simplify: (x^2)’ = 2x^1

4. Rewrite the equation as y’ = 2x

Question:

2/(x+1)−x^2/(3x−1)

Answer:

1. Simplify the left side: 2/(x+1)
2. Simplify the right side: -x^2/(3x-1)
3. Combine the two sides: 2/(x+1) - (-x^2/(3x-1))
4. Combine the denominators: (3x-1)(x+1)
5. Multiply both sides by the denominator: (3x-1)(x+1) [2/(x+1) - (-x^2/(3x-1))]
6. Simplify the left side: (3x-1) + (2x^2)
7. Simplify the right side: -x^3 - 2
8. Combine the two sides: (3x-1) + (2x^2) - (-x^3 - 2)
9. Combine like terms: 3x-1 + 2x^2 + x^3 + 2
10. Simplify: 4x^3 + 3x - 1

Question:

Find the derivative of cosx by first principle.

Answer:

Answer: Step 1: Let y = cosx

Step 2: Calculate the derivative of y with respect to x using the first principle:

d/dx(y) = limh→0 (y + h - y) / h

Step 3: Substitute y = cosx into the equation:

d/dx(cosx) = limh→0 (cosx + h - cosx) / h

Step 4: Simplify the equation:

d/dx(cosx) = limh→0 (h) / h

Step 5: The limit of h as it approaches 0 is 0, so the equation becomes:

d/dx(cosx) = 0

Question:

Find the derivative of 99x at x=100

Answer:

Answer: Step 1: Rewrite the equation as y = 99x

Step 2: Take the derivative of y with respect to x: dy/dx = 99

Step 3: Evaluate the derivative at x=100: dy/dx = 99(100) = 9900

Question:

Find the derivatives of the following: 3cotx+5cosecx.

Answer:

Answer:

Step 1: Use the chain rule to find the derivative of 3cotx.

d/dx (3cotx) = -3csc2x

Step 2: Use the chain rule to find the derivative of 5cosecx.

d/dx (5cosecx) = 5cosecxcotx

Therefore, the derivatives of 3cotx+5cosecx is -3csc2x + 5cosecxcotx.

Question:

Find the derivative of (x^n−a^n)/(x−a) for some constant a

Answer:

Answer:

1. Rewrite the given expression as: (x^n - a^n)/(x - a)

2. Apply the quotient rule for derivatives: n*(x^(n-1))(x - a) - n(a^(n-1))/(x - a)^2

3. Simplify the fraction: n*(x^(n-1)) - n*(a^(n-1))/(x - a)

4. Factor out the common terms: n*(x^(n-1) - a^(n-1))/(x - a)