Differentiate following with respect to x f(x) = (3x^2 – 5ax + a^2)^4
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To differentiate the given function f(x) = (3x^2 – 5ax + a^2)^4 with respect to x, we can use the chain rule. The chain rule states that if we have a composite function u(v(x)), then its derivative with respect to x is given by du/dx = du/dv * dv/dx.
In this case, let's define u as u = (3x^2 – 5ax + a^2), and consider f(x) as f(u) = u^4. To differentiate f(x) with respect to x, we will differentiate f(u) with respect to u, and then multiply it by du/dx.
1. Differentiate f(u) = u^4 with respect to u:
To differentiate u^4 with respect to u, we can simply bring down the power to multiply, and then differentiate u with respect to u (which is 1). So, df/du = 4u^3.
2. Calculate du/dx:
To find du/dx, we need to differentiate u = (3x^2 – 5ax + a^2) with respect to x. To do this, we will differentiate each term separately using the power rule and the constant multiple rule.
- Differentiate 3x^2: The derivative of 3x^2 with respect to x is 6x.
- Differentiate -5ax: The derivative of -5ax with respect to x is -5a.
- Differentiate a^2: Since a^2 is a constant, the derivative is 0.
Combining these results, du/dx = 6x - 5a.
3. Multiply df/du by du/dx:
Multiply the derivative df/du = 4u^3 by du/dx = 6x - 5a.
So, df/dx = (6x - 5a)(4u^3).
Substituting the value of u back in, we have:
df/dx = (6x - 5a)(4(3x^2 – 5ax + a^2)^3).
Therefore, the derivative of f(x) = (3x^2 – 5ax + a^2)^4 with respect to x is (6x - 5a)(4(3x^2 – 5ax + a^2)^3).