In the realm of calculus, the derivative signifies an immediate rate of alteration. Essentially, it gauges how the output of a function varies in response to changes in its input. There are numerous methods to ascertain derivatives, such as the power rule, product rule, quotient rule, and chain rule. These rules are crucial to numerous fields, including mathematics and physics.
| Topic | Problem | Solution |
|---|---|---|
| None | Find equations of the tangent line and normal lin… | Given the equation for y as a function of x, we have \(y = 14\cos(x)\). |
| None | Find $f_{x}$ and $f_{y}$ for $f(x, y)=y \ln (9 x+… | Given the function \(f(x, y)=y \ln (9 x+y)\), we need to find the partial derivatives \(f_x\) and \… |
| None | (a) Differentiate $A(m, h)$ with respect to $m$ t… | Differentiate the function $A(m, h) = 0.024265 h^{0.3964} m^{0.5378}$ with respect to $m$ using the… |
| None | Find $f_{x}, f_{y}, f_{x}(-4,1)$, and $f_{y}(-1,-… | Given the function \(f(x, y)=\sqrt{x^{2}+y^{2}}\), we are asked to find the partial derivatives of … |
| None | Given the function $g(x)=4 x^{3}+6 x^{2}-24 x$, f… | The derivative of a function can be found by applying the power rule, which states that the derivat… |
| None | Find $\frac{d r}{d \theta}$. \[ r=4-\theta^{4} \s… | We are given the function \(r = 4 - \theta^4 \sin \theta\) and we are asked to find \(\frac{d r}{d … |
| None | Find $\frac{d s}{d t}$ \[ s=2 \cot t-e^{-t} \] | Given the function \(s=2 \cot t-e^{-t}\). |
| None | Find $\frac{d y}{d x}$ \[ y=\cot x-6 \sqrt{x}+\fr… | First, we need to find the derivative of each term in the equation separately. |
| None | Differentiate the function and find the slope of … | Given the function \(s=t^{3}-t^{2}\) and \(t=-6\) |