Find the derivatives of the following functions using the chain rule: y = (2x3 - 3x + 5)4, y = (3x3 + 7)3, y…

Question

Find the derivatives of the following functions using the chain rule: y = (2x3 - 3x + 5)4, y = (3x3 + 7)3, y = (x4 - 4)4.

ScanSolve AI Tutor · Accepted Answer

Here are the derivatives of the given functions:

1. $y = (2x^3 - 3x + 5)^4$

Apply the chain rule, $\frac{dy}{dx} = n[f(x)]^{n-1} \cdot f'(x)$ .
Here, $f(x) = 2x^3 - 3x + 5$ and $n=4$ .
$\frac{dy}{dx} = 4(2x^3 - 3x + 5)^{4-1} \cdot \frac{d}{dx}(2x^3 - 3x + 5)$
Differentiate the inner function.
$\frac{d}{dx}(2x^3 - 3x + 5) = 6x^2 - 3$
Substitute the derivative of the inner function back into the chain rule expression.
$\frac{dy}{dx} = 4(2x^3 - 3x + 5)^3 (6x^2 - 3)$ $\frac{dy}{dx} = 12(2x^3 - 3x + 5)^3 (2x^2 - 1)$
The derivative is $\boxed{12(2x^3 - 3x + 5)^3 (2x^2 - 1)}$ .

2. $y = (3x^3 + 7)^3$
Apply the chain rule.
Here, $f(x) = 3x^3 + 7$ and $n=3$ .
$\frac{dy}{dx} = 3(3x^3 + 7)^{3-1} \cdot \frac{d}{dx}(3x^3 + 7)$
Differentiate the inner function.
$\frac{d}{dx}(3x^3 + 7) = 9x^2$
Substitute the derivative of the inner function.
$\frac{dy}{dx} = 3(3x^3 + 7)^2 (9x^2)$ $\frac{dy}{dx} = 27x^2(3x^3 + 7)^2$
The derivative is $\boxed{27x^2(3x^3 + 7)^2}$ .

3. $y = (x^4 - 4)^4$
Apply the chain rule.
Here, $f(x) = x^4 - 4$ and $n=4$ .
$\frac{dy}{dx} = 4(x^4 - 4)^{4-1} \cdot \frac{d}{dx}(x^4 - 4)$
Differentiate the inner function.
$\frac{d}{dx}(x^4 - 4) = 4x^3$
Substitute the derivative of the inner function.
$\frac{dy}{dx} = 4(x^4 - 4)^3 (4x^3)$ $\frac{dy}{dx} = 16x^3(x^4 - 4)^3$
The derivative is $\boxed{16x^3(x^4 - 4)^3}$ .

4. $y = (2x - 1)^2 (3x^2 + 1)^2$
Apply the product rule, $\frac{dy}{dx} = u'v + uv'$ .
Let $u = (2x - 1)^2$ and $v = (3x^2 + 1)^2$ .
Find $u'$ using the chain rule.
$u' = \frac{d}{dx}(2x - 1)^2 = 2(2x - 1)^{2-1} \cdot \frac{d}{dx}(2x - 1) = 2(2x - 1)(2) = 4(2x - 1)$
Find $v'$ using the chain rule.
$v' = \frac{d}{dx}(3x^2 + 1)^2 = 2(3x^2 + 1)^{2-1} \cdot \frac{d}{dx}(3x^2 + 1) = 2(3x^2 + 1)(6x) = 12x(3x^2 + 1)$
Substitute $u, u', v, v'$ into the product rule formula.
$\frac{dy}{dx} = [4(2x - 1)](3x^2 + 1)^2 + (2x - 1)^2 [12x(3x^2 + 1)]$
Factor out common terms.
The common terms are $(2x-1)$ and $(3x^2+1)$ .
$\frac{dy}{dx} = (2x - 1)(3x^2 + 1) [4(3x^2 + 1) + 12x(2x - 1)]$
Expand and simplify the terms inside the square brackets.
$\frac{dy}{dx} = (2x - 1)(3x^2 + 1) [12x^2 + 4 + 24x^2 - 12x]$ $\frac{dy}{dx} = (2x - 1)(3x^2 + 1) [36x^2 - 12x + 4]$
Factor out 4 from the bracketed term.
$\frac{dy}{dx} = 4(2x - 1)(3x^2 + 1) (9x^2 - 3x + 1)$
The derivative is $\boxed{4(2x - 1)(3x^2 + 1) (9x^2 - 3x + 1)}$ .

Find the derivatives of the following functions using the chain rule: y = (2x3 − 3x + 5)4, y = (3x3 + 7)3, y = (x4 − 4)4.

Step-by-step solution

Apply the chain rule, $\frac{dy}{dx} = n[f(x)]^{n-1} \cdot f'(x)$ .

Differentiate the inner function.

Substitute the derivative of the inner function back into the chain rule expression.

Apply the chain rule.

Differentiate the inner function.

Substitute the derivative of the inner function.

Apply the chain rule.

Differentiate the inner function.

Substitute the derivative of the inner function.

Apply the product rule, $\frac{dy}{dx} = u'v + uv'$ .

Find $u'$ using the chain rule.

Find $v'$ using the chain rule.

Substitute $u, u', v, v'$ into the product rule formula.

Factor out common terms.

Expand and simplify the terms inside the square brackets.

Factor out 4 from the bracketed term.

Still stuck on this one?

More Mathematics Questions

Find the derivatives of the following functions using the chain rule: y = (2x3 − 3x + 5)4, y = (3x3 + 7)3, y = (x4 − 4)4.

Step-by-step solution

Apply the chain rule, dydx=n[f(x)]n−1⋅f′(x)\frac{dy}{dx} = n[f(x)]^{n-1} \cdot f'(x)dxdy​=n[f(x)]n−1⋅f′(x).

Differentiate the inner function.

Substitute the derivative of the inner function back into the chain rule expression.

Apply the chain rule.

Differentiate the inner function.

Substitute the derivative of the inner function.

Apply the chain rule.

Differentiate the inner function.

Substitute the derivative of the inner function.

Apply the product rule, dydx=u′v+uv′\frac{dy}{dx} = u'v + uv'dxdy​=u′v+uv′.

Find u′u'u′ using the chain rule.

Find v′v'v′ using the chain rule.

Substitute $u, u', v, v'$ into the product rule formula.

Factor out common terms.

Expand and simplify the terms inside the square brackets.

Factor out 4 from the bracketed term.

Still stuck on this one?

More Mathematics Questions

Apply the chain rule, $\frac{dy}{dx} = n[f(x)]^{n-1} \cdot f'(x)$ .

Apply the product rule, $\frac{dy}{dx} = u'v + uv'$ .

Find $u'$ using the chain rule.

Find $v'$ using the chain rule.