In the sphere of calculus, the Chain Rule emerges as a practical approach for determining the derivative of composite functions. This principle suggests that the derivative of a composite function equals the derivative of the exterior function times the derivative of the interior function. It serves as an indispensable instrument for differentiating functions that aren't directly manageable.
| Topic | Problem | Solution |
|---|---|---|
| None | Calculate the derivative of the following functio… | The derivative of a function can be calculated using the chain rule. The chain rule states that the… |
| None | Find the derivative. Find $f^{\prime}(x)$ for $f(… | Given the function \(f(x)=(4 x^{2}+3 x)^{2}\) |
| None | unction $y=\left(\frac{3 x^{2}-5}{2 x+1}\right)^{… | The problem is asking for the derivative of the function \(y=\left(\frac{3 x^{2}-5}{2 x+1}\right)^{… |
| None | Find the derivative of the function. \[ \begin{ar… | The derivative of a function can be found using the rules of differentiation. In this case, we have… |
| None | Find the derivative of the function. \[ f(x)=e^{2… | The function given is \(f(x)=e^{2-x}\). |
| None | 1. What is the derivative of $\cos ^{2} 3 x$ with… | We are given the function \(f = \cos^2(3x)\) and we are asked to find its derivative with respect t… |
| None | Use the Chain Rule to differentiate the following… | Given the function \(f(x)=\sqrt{x^{2}+\sqrt{2-5 x^{3}}}\) |
| None | Find the derivative of the function $z=\left(t e^… | Given the function \(z=(t e^{8 t}+e^{7 t})^{9}\) |
| None | Find the derivative of the given function. \[ q=\… | We are given the function \(q=\sin \left(\frac{t}{\sqrt{t+2}}\right)\) and we are asked to find its… |
| None | Use part I of the Fundamental Theorem of Calculus… | The Fundamental Theorem of Calculus Part I states that if a function f is continuous on the interva… |
| None | Suppose $f^{\prime}(x)=\sin \left(5 x^{2}\right)$… | Given that the derivative of the function $f(x)$ is $f'(x) = \sin(5x^2)$ |
| None | 2. If $p=\frac{q^{2}}{\pi}, q=3 \cos (r)+\sin (r)… | First, substitute the expression for r into the expression for q: \(q = 3 \cos (\ln (s^\pi)) + \sin… |
| None | Find the derivative using the chain rule. \[ f(k)… | Find the derivative of the function using the chain rule: \(f'(k) = \frac{1}{2\sqrt{2k^9 + 7}} \cdo… |
| None | Find the derivative using the chain rule. \[ f(a)… | Find the derivative of the outer function: \(\frac{1}{2\sqrt{8a^7 + 6}}\) |
| None | 13. Compute the derivatives of the following func… | \( \frac{d}{dx}(6 e^{x^{4}+x^{2}}) = 6 \frac{d}{dx} (e^{x^{4}+x^{2}})\) |