Find the dy/dx for each of the following
1)y=ln(2x^6-15)
2)y= 3√ 2x^2-3x-1/5
3) y= x^4/x^2+1
1. dy/dx = 12x^5 /(2x^6 - 15) ---- routine
2. dy/dx = 6√2 x - 3 ---- routine
3. the way you typed it,
y = x^4/x^2 + 1
y = x^2 + 1
dy/dx = 2x
if you meant x^4/(x^2+ 1) , then you would use the quotient rule. See how critical it is to use brackets ?
there is no brackets for question 3. Although i am still confused. would you be able to give step by step solutions?
To find the derivatives of the given functions, we will use the power rule, chain rule, and quotient rule. Here's how to find dy/dx for each function:
1) y = ln(2x^6 - 15)
To find dy/dx, we will use the chain rule. The chain rule states that if you have a composite function, f(g(x)), then the derivative can be found by taking the derivative of the outer function multiplied by the derivative of the inner function.
Let g(x) = 2x^6 - 15 (the inner function), and f(x) = ln(x) (the outer function).
Using the chain rule, dy/dx = f'(g(x)) * g'(x) = (1/g(x)) * g'(x).
Now, let's find g'(x):
g'(x) = d/dx(2x^6 - 15) = 12x^5.
Substituting this into the chain rule formula:
dy/dx = (1 / (2x^6 - 15)) * 12x^5.
2) y = 3√ (2x^2 - 3x - 1)/5
To find dy/dx, we will use the chain rule and the power rule. The power rule states that if y = x^n, then dy/dx = n*x^(n-1).
Let's rewrite the function as:
y = 3(2x^2 - 3x - 1)^(1/5) / 5.
Using the chain rule, let g(x) = 2x^2 - 3x - 1 (the inner function), and f(x) = x^(1/5) (the outer function).
Now, let's find g'(x):
g'(x) = d/dx(2x^2 - 3x - 1) = 4x - 3.
Substituting this into the chain rule formula:
dy/dx = (1/5) * (3(2x^2 - 3x - 1))^(-4/5) * (4x - 3).
3) y = x^4 / (x^2 + 1)
To find dy/dx, we will use the quotient rule. The quotient rule states that if y = f(x)/g(x), then dy/dx = (g(x)*f'(x) - f(x)*g'(x)) / (g(x))^2.
Let f(x) = x^4 and g(x) = x^2 + 1.
Now, let's find f'(x) and g'(x):
f'(x) = d/dx(x^4) = 4x^3,
g'(x) = d/dx(x^2 + 1) = 2x.
Substituting these into the quotient rule formula:
dy/dx = ((x^2 + 1)*(4x^3) - (x^4)*(2x)) / (x^2 + 1)^2.
Simplifying the expression:
dy/dx = (4x^5 + 4x^3 - 2x^5) / (x^2 + 1)^2.
Therefore, the derivatives for the given functions are:
1) dy/dx = (1 / (2x^6 - 15)) * 12x^5
2) dy/dx = (1/5) * (3(2x^2 - 3x - 1))^(-4/5) * (4x - 3)
3) dy/dx = (4x^5 + 4x^3 - 2x^5) / (x^2 + 1)^2.