Can anyone explain how to use the chain rule and power rule together to find the first derivative and please do these examples.
f(x) = -3x^4/(sqrt(4x-8))
g(x) = ((2x+5)/(6-x^2)^4
g(x) = (8x^3)(4x^2+2x-3)^5
y = [1/(4x+x^2)^3]^3
Thank you so much!
I will give you the 'first-line' derivatives of the first two questions
a) f'(x) = [(4x-8)^(1/2)(-12x^3) - (-3x^4)(1/2(4x-8)^(-1/2)(4)]/(4x-8)
= ....
b) I see you had a double bracket at the front, but one bracket is not closed.
Did you mean [(2x+5)/(6-x^2)]^4 ?
c) g(x) = (8x^3)(4x^2+2x-3)^5
g'(x) = 5(4x^2+2x-3)^5(8x+2)(8x^3) + 24x^2(4x^2+2x-3)^5
= ....
d) y = [1/(4x+x^2)^3]^3
y = (4x+x^2)^-9
y' = -9(4x + x^2)^-10(4+2x)
= ....
To use the chain rule and power rule together, follow these steps:
1. Identify the outer function and the inner function in the given expression.
2. Apply the power rule to differentiate the outer function, treating the inner function as a constant.
3. Apply the chain rule to differentiate the inner function, treating the outer function as a constant.
4. Multiply the results obtained from steps 2 and 3 to find the total derivative.
Now, let's apply these steps to the given examples:
Example 1:
f(x) = -3x^4/(sqrt(4x-8))
Step 1:
The outer function is the division, and the inner function is the square root.
Step 2:
Differentiating the outer function:
f'(x) = (-3x^4)' * (1/sqrt(4x-8)) - (3x^4) * (sqrt(4x-8))'
Step 3:
Differentiating the inner function:
(1/sqrt(4x-8)) = (4x-8)^(-1/2)
(4x-8)^(-1/2)' = (-1/2)*(4x-8)^(-3/2)*(4-0)
Step 4:
f'(x) = -12x^3/(sqrt(4x-8)) + (3x^4)(1/2)(4x-8)^(-3/2)*(4-0)
= -12x^3/(sqrt(4x-8)) + 6x^4(4x-8)^(-3/2)
Example 2:
g(x) = ((2x+5)/(6-x^2))^4
Step 1:
The outer function is the power function, and the inner function is the division.
Step 2:
Differentiating the outer function:
g'(x) = (4)((2x+5)/(6-x^2))^3 * ((2x+5)/(6-x^2))'
Step 3:
Differentiating the inner function:
((2x+5)/(6-x^2))' = ((2)(6-x^2)-(2x+5)(-2x))/((6-x^2))^2
Step 4:
g'(x) = (4)((2x+5)/(6-x^2))^3 * ((2)(6-x^2)-(2x+5)(-2x))/((6-x^2))^2
Example 3:
g(x) = (8x^3)(4x^2+2x-3)^5
Step 1:
The outer function is the multiplication, and the inner function is the power function.
Step 2:
Differentiating the outer function:
g'(x) = (8)((4x^2+2x-3)^5)' + (8x^3)(4x^2+2x-3)^5'
Step 3:
Differentiating the inner function:
(4x^2+2x-3)^5' = 5(4x^2+2x-3)^4 * (4x^2+2x-3)'
Step 4:
g'(x) = (8)(4x^2+2x-3)^5 + (8x^3)(5)(4x^2+2x-3)^4 * (4x^2+2x-3)'
Example 4:
y = [1/(4x+x^2)^3]^3
Step 1:
The outer function is the power function, and the inner function is the reciprocal.
Step 2:
Differentiating the outer function:
y' = (3)[1/(4x+x^2)^3]^2 * [(1/(4x+x^2)^3)']
Step 3:
Differentiating the inner function:
(1/(4x+x^2)^3)' = -3(4x+x^2)^2(4+2x)
Step 4:
y' = (3)[1/(4x+x^2)^3]^2 * [-3(4x+x^2)^2(4+2x)]
These are the step-by-step calculations and explanations for finding the first derivatives using the chain rule and power rule.
To use the chain rule and power rule together to find the first derivative, we need to follow these steps for each example:
1. Identify the composite function within the equation, denoted as u(x).
2. Take the derivative of the composite function, u'(x), using the power rule.
3. Determine the derivative of the outer function with respect to u, denoted as dy/du.
4. Multiply the derivatives from steps 2 and 3 together to obtain the derivative of the entire expression, dy/dx.
Now, let's apply these steps to the given examples:
Example 1: f(x) = -3x^4 / sqrt(4x-8)
Step 1: Identify the composite function: u(x) = 4x - 8 (inside the square root).
Step 2: Take the derivative of the composite function:
u'(x) = d(4x - 8)/dx = 4
Step 3: Determine the derivative of the outer function with respect to u:
dy/du = d(-3x^4)/du = -12x^3
Step 4: Multiply the derivatives:
dy/dx = dy/du * du/dx = -12x^3 * 4 = -48x^3
Therefore, the first derivative of f(x) is dy/dx = -48x^3.
Example 2: g(x) = ((2x+5)/(6-x^2))^4
Step 1: Identify the composite function: u(x) = 6 - x^2 (in the denominator).
Step 2: Take the derivative of the composite function:
u'(x) = d(6 - x^2)/dx = -2x
Step 3: Determine the derivative of the outer function with respect to u:
dy/du = d((2x + 5)^4)/du = 4(2x + 5)^3
Step 4: Multiply the derivatives:
dy/dx = dy/du * du/dx = 4(2x + 5)^3 * (-2x) = -8x(2x + 5)^3
Therefore, the first derivative of g(x) is dy/dx = -8x(2x + 5)^3.
Example 3: g(x) = (8x^3)(4x^2 + 2x - 3)^5
Step 1: Identify the composite function: u(x) = 4x^2 + 2x - 3 (inside the parentheses).
Step 2: Take the derivative of the composite function:
u'(x) = d(4x^2 + 2x - 3)/dx = 8x + 2
Step 3: Determine the derivative of the outer function with respect to u:
dy/du = d(8x^3)/du = 24x^2
Step 4: Multiply the derivatives:
dy/dx = dy/du * du/dx = 24x^2 * (8x + 2) = 192x^3 + 48x^2
Therefore, the first derivative of g(x) is dy/dx = 192x^3 + 48x^2.
Example 4: y = [1/(4x + x^2)^3]^3
Step 1: Identify the composite function: u(x) = 4x + x^2 (in the denominator).
Step 2: Take the derivative of the composite function:
u'(x) = d(4x + x^2)/dx = 4 + 2x
Step 3: Determine the derivative of the outer function with respect to u:
dy/du = d([1/u]^3)/du = -3/u^4
Step 4: Multiply the derivatives:
dy/dx = dy/du * du/dx = -3/u^4 * (4 + 2x)
Replacing u with (4x + x^2):
dy/dx = -3/(4x + x^2)^4 * (4 + 2x)
Therefore, the first derivative of y is dy/dx = -3/(4x + x^2)^4 * (4 + 2x).
In summary, we can use the chain rule and power rule together to find the first derivative of a function by first identifying the composite functions and then taking the derivatives of both the composite and outer functions. Finally, we multiply these derivatives together to obtain the first derivative of the entire expression.