here are some important B.Sc. IT Semester 5 questions on successive differentiation along with step-by-step solutions. These problems are designed to help students understand the concepts thoroughly.
Problem 1: Successive Differentiation of a Polynomial
Problem: Find the first, second, and third derivatives of the function f(x)=4x3−3x2+2x−5.
Solution:
First Derivative:
- Write the function: f(x)=4x3−3x2+2x−5
- Differentiate each term:
- The derivative of 4x3 is 12x2
- The derivative of −3x2 is −6x
- The derivative of 2x is 2
- The derivative of −5 is 0 (since the derivative of a constant is zero)
- Combine the results:
f′(x)=12x2−6x+2
Second Derivative:
- Write the first derivative: f′(x)=12x2−6x+2
- Differentiate each term:
- The derivative of 12x2 is 24x
- The derivative of −6x is −6
- The derivative of 2 is 0
- Combine the results:
f′′(x)=24x−6
Third Derivative:
- Write the second derivative: f′′(x)=24x−6
- Differentiate each term:
- The derivative of 24x is 24
- The derivative of −6 is 0
- Combine the results:
f′′′(x)=24
Problem 2: Successive Differentiation of an Exponential Function
Problem: Find the first and second derivatives of the function g(x)=e4x.
Solution:
First Derivative:
- Write the function: g(x)=e4x
- Use the chain rule. The outer function is eu where u=4x:
- The derivative of eu is eu
- Multiply by the derivative of u: dxd(4x)=4
- Combine the results:
g′(x)=e4x⋅4=4e4x
Second Derivative:
- Write the first derivative: g′(x)=4e4x
- Use the chain rule again. The outer function is eu where u=4x:
- The derivative of eu is eu
- Multiply by the derivative of u: dxd(4x)=4
- Combine the results:
g′′(x)=4⋅e4x⋅4=16e4x
Problem 3: Successive Differentiation of a Trigonometric Function
Problem: Find the first, second, and third derivatives of the function h(x)=sin(2x).
Solution:
First Derivative:
- Write the function: h(x)=sin(2x)
- Use the chain rule. The outer function is sin(u) where u=2x:
- The derivative of sin(u) is cos(u)
- Multiply by the derivative of u: dxd(2x)=2
- Combine the results:
h′(x)=cos(2x)⋅2=2cos(2x)
Second Derivative:
- Write the first derivative: h′(x)=2cos(2x)
- Use the chain rule again. The outer function is cos(u) where u=2x:
- The derivative of cos(u) is −sin(u)
- Multiply by the derivative of u: dxd(2x)=2
- Combine the results:
h′′(x)=2⋅−sin(2x)⋅2=−4sin(2x)
Third Derivative:
- Write the second derivative: h′′(x)=−4sin(2x)
- Use the chain rule again. The outer function is sin(u) where u=2x:
- The derivative of sin(u) is cos(u)
- Multiply by the derivative of u: dxd(2x)=2
- Combine the results:
h′′′(x)=−4⋅cos(2x)⋅2=−8cos(2x)
Problem 4: Successive Differentiation of a Logarithmic Function
Problem: Find the first and second derivatives of the function k(x)=ln(2x).
Solution:
First Derivative:
- Write the function: k(x)=ln(2x)
- Use the chain rule. The outer function is ln(u) where u=2x:
- The derivative of ln(u) is u1
- Multiply by the derivative of u: dxd(2x)=2
- Combine the results:
k′(x)=2x1⋅2=2x2=x1
Second Derivative:
- Write the first derivative: k′(x)=x1
- Rewrite x1 as x−1
- The derivative of x−1 is −x−2
- Combine the results:
k′′(x)=−x21
Problem 5: Successive Differentiation of a Rational Function
Problem: Find the first and second derivatives of the function m(x)=x+21.
Solution:
First Derivative:
- Write the function: m(x)=x+21
- Rewrite x+21 as (x+2)−1
- Use the power rule and chain rule:
- The derivative of (x+2)−1 is −1⋅(x+2)−2
- Multiply by the derivative of x+2, which is 1
- Combine the results:
m′(x)=−(x+2)21
Second Derivative:
- Write the first derivative: m′(x)=−(x+2)21
- Rewrite −(x+2)21 as −(x+2)−2
- Use the power rule and chain rule again:
- The derivative of −(x+2)−2 is 2(x+2)−3
- Multiply by the derivative of x+2, which is 1
- Combine the results:
m′′(x)=(x+2)32
Problem 6: Successive Differentiation of a Combined Function
Problem: Find the first and second derivatives of the function n(x)=x2⋅ln(x).
Solution:
First Derivative:
- Write the function: n(x)=x2⋅ln(x)
- Use the product rule: (uv)′=u′v+uv′
- Let u=x2 and v=ln(x)
- The derivative of u=x2 is u′=2x
- The derivative of v=ln(x) is v′=x1
- Combine the results:
n′(x)=(2x)⋅ln(x)+(x2)⋅x1=2xln(x)+x
Second Derivative:
- Write the first derivative: n′(x)=2xln(x)+x
- Differentiate each term using the product rule again for 2xln(x):
- Let u=2x and v=ln(x)
- The derivative of u=2x is u′=2
- The derivative of v=ln(x) is v′=x1
- Combine the results for the first term:
(2xln(x))′=(2)⋅ln(x)+(2x)⋅x1=2ln(x)+2
- The derivative of the second term x is 1
- Combine all results:
n′′(x)=2ln(x)+2+1=2ln(x)+3
Problem 7: Successive Differentiation of a Trigonometric Function
Problem: Find the first, second, and third derivatives of the function f(x)=cos(3x).
Solution:
First Derivative:
- Write the function: f(x)=cos(3x)
- Use the chain rule. The outer function is cos(u) where u=3x:
- The derivative of cos(u) is −sin(u)
- Multiply by the derivative of u: dxd(3x)=3
- Combine the results:
f′(x)=−sin(3x)⋅3=−3sin(3x)
Second Derivative:
- Write the first derivative: f′(x)=−3sin(3x)
- Use the chain rule again. The outer function is sin(u) where u=3x:
- The derivative of sin(u) is cos(u)
- Multiply by the derivative of u: dxd(3x)=3
- Combine the results:
f′′(x)=−3⋅cos(3x)⋅3=−9cos(3x)
Third Derivative:
- Write the second derivative: f′′(x)=−9cos(3x)
- Use the chain rule again. The outer function is cos(u) where u=3x:
- The derivative of cos(u) is −sin(u)
- Multiply by the derivative of u: dxd(3x)=3
- Combine the results:
f′′′(x)=−9⋅−sin(3x)⋅3=27sin(3x)
Problem 8: Successive Differentiation of a Logarithmic Function
Problem: Find the first and second derivatives of the function g(x)=ln(5x).
Solution:
First Derivative:
- Write the function: g(x)=ln(5x)
- Use the chain rule. The outer function is ln(u) where u=5x:
- The derivative of ln(u) is u1
- Multiply by the derivative of u: dxd(5x)=5
- Combine the results:
g′(x)=5x1⋅5=5x5=x1
Second Derivative:
- Write the first derivative: g′(x)=x1
- Rewrite x1 as x−1
- The derivative of x−1 is −x−2
- Combine the results:
g′′(x)=−x21
Problem 9: Successive Differentiation of a Polynomial Function
Problem: Find the first, second, and third derivatives of the function h(x)=2x4−x3+3x−7.
Solution:
First Derivative:
- Write the function: h(x)=2x4−x3+3x−7
- Differentiate each term:
- The derivative of 2x4 is 8x3
- The derivative of −x3 is −3x2
- The derivative of 3x is 3
- The derivative of −7 is 0
- Combine the results:
h′(x)=8x3−3x2+3
Second Derivative:
- Write the first derivative: h′(x)=8x3−3x2+3
- Differentiate each term:
- The derivative of 8x3 is 24x2
- The derivative of −3x2 is −6x
- The derivative of 3 is 0
- Combine the results:
h′′(x)=24x2−6x
Third Derivative:
- Write the second derivative: h′′(x)=24x2−6x
- Differentiate each term:
- The derivative of 24x2 is 48x
- The derivative of −6x is −6
- Combine the results:
h′′′(x)=48x−6
Problem 10: Successive Differentiation of an Exponential Function
Problem: Find the first and second derivatives of the function k(x)=e2x.
Solution:
First Derivative:
- Write the function: k(x)=e2x
- Use the chain rule. The outer function is eu where u=2x:
- The derivative of eu is eu
- Multiply by the derivative of u: dxd(2x)=2
- Combine the results:
k′(x)=e2x⋅2=2e2x
Second Derivative:
- Write the first derivative: k′(x)=2e2x
- Use the chain rule again. The outer function is eu where u=2x:
- The derivative of eu is eu
- Multiply by the derivative of u: dxd(2x)=2
- Combine the results:
k′′(x)=2⋅e2x⋅2=4e2x
Problem 11: Successive Differentiation of a Rational Function
Problem: Find the first and second derivatives of the function m(x)=2x+31.
Solution:
First Derivative:
- Write the function: m(x)=2x+31
- Rewrite 2x+31 as (2x+3)−1
- Use the power rule and chain rule:
- The derivative of (2x+3)−1 is −1⋅(2x+3)−2
- Multiply by the derivative of 2x+3, which is 2
- Combine the results:
m′(x)=−(2x+3)21⋅2=−(2x+3)22
Second Derivative:
- Write the first derivative: m′(x)=−(2x+3)22
- Rewrite −(2x+3)22 as −2(2x+3)−2
- Use the power rule and chain rule again:
- The derivative of −2(2x+3)−2 is −2⋅−2⋅(2x+3)−3
- Multiply by the derivative of 2x+3, which is 2
- Combine the results:
m′′(x)=4⋅(2x+3)31=(2x+3)38
Problem 12: Successive Differentiation of a Combined Function
Problem: Find the first and second derivatives of the function n(x)=x3⋅ex.
Solution:
First Derivative:
- Write the function: n(x)=x3⋅ex
- Use the product rule: (uv)′=u′v+uv′
- Let u=x3 and v=ex
- The derivative of u=x3 is u′=3x2
- The derivative of v=ex is v′=ex
- Combine the results:
n′(x)=(3x2)⋅ex+(x3)⋅ex=3x2ex+x3ex
- Factor out the common term ex:
n′(x)=ex(3x2+x3)=exx2(3+x)
Second Derivative:
- Write the first derivative: n′(x)=ex(3x2+x3)
- Use the product rule again:
- Let u=ex and v=3x2+x3
- The derivative of u=ex is u′=ex
- The derivative of v=3x2+x3 is v′=6x+3x2
- Combine the results:
n′′(x)=(ex)⋅(6x+3x2)+(ex)⋅(3x2+x3)=ex(6x+3x2+3x2+x3)=ex(6x+6x2+x3)
- Simplify the expression:
n′′(x)=ex(x3+6x2+6x)