Finding a Tangent Line to a Curve Introduction,Examples with Step-by-Step Explanations

The problems about Finding a Tangent Line to a Curve

Topic	Problem	Solution
None	Use implicit differentiation to find $y^{'}$ …	Differentiate the given equation implicitly with respect to x. This involves applying the product r…
None	Use implicit differentiation to find $d y / d x$ .…	Use implicit differentiation to find $\frac{d y}{d x}$ .
None	Find an equation of the line tangent to the graph…	First, we need to find the derivative of the function $f (x) = 2 x^{3}$ . The derivative of a function…
None	Assuming that the equations define $x$ and $y$ im…	Given the equations $x = t^{5} + t$ and $y + 4 t^{5} = 4 x + t^{3}$ at $t = 2$ , we are asked to find the …
None	Find the equation of the tangent line at the give…	First, we need to find the derivative of the given function. The derivative of \(3y^2 - \sqrt{x} = …
None	Find an equation of the tangent line to the given…	Given the function $y = \frac{e^{8 x}}{x}$ and the point $(\frac{1}{8}, 8 e)$ , we are …
None	Find the linearization $L (x, y)$ of the function …	Given the function $f (x, y) = x^{2} + y^{2} + 1$ , we want to find its linearization at the point (4,1…
None	Find the equation of the line tangent to the grap…	Let's find the derivative of the function $f (x) = (\ln x)^{4}$ .
None	possible The given point is on the curve. Find th…	We are given the curve $6 x^{2} + 7 x y + 2 y^{2} + 13 y - 6 = 0$ and the point $(- 1, 0)$ . We are asked to…
None	Find an equation for the tangent to the curve at …	Given the function $f (x) = 10 \sqrt{x} - x + 7$ and the point $(100, 7)$ .
None	Find an equation for the tangent plane to the sur…	Given the surface equation $z + 5 = x y^{3} \cos (z)$ , we want to find the equation of the tangent plane a…
None	Find a linearization that will replace the functi…	We are given the function $f (x) = \sqrt[4]{x}$ and we are asked to find a linearization of this f…
None	Let $f (x) = 6 x^{2}$ a) Find the linearization $L(x…	The function given is $f (x) = 6 x^{2}$ .
None	Find the slope of the tangent line to the curve \…	First, we need to find the derivative of the given equation. The derivative of a function gives us …
None	Assuming that the equation defines $x$ and $y$ im…	Given the equations $x^{3} + 4 t^{2} = 65$ and $2 y^{3} - 3 t^{2} = 80$ , we are asked to find the slope…
None	Find (a) the slope of the curve at the given poin…	Given the function $y = - 2 - 4 x^{2}$ and the point $P (3, - 38)$ .
None	a. Find an equation of the tangent line at $x = a$ .…	We are given the function $y = e^{x}$ and $a = \ln 10$ .
None	Find an equation of the tangent line to the graph…	Find the derivative of the function using the chain rule: $\frac{d y}{d x} = - 2 x e^{- x^{2}}$
None	The line with equation $y = - 2 x + c$ is tangent to t…	Rearrange the equation of the line: $y = - 2 x + c$ to get $- 2 x = y - c$
None	The function $f$ is given by $f(x)=\frac{1}{3} e^…	$f (x) = \frac{1}{3} e^{- 3 x} + 1$
None	For the function $f (x) = x^{2} + 2$ , find the equatio…	First, find the slope of the tangent line by taking the derivative of the function: $f^{'} (x) = 2 x$
None	15 Point $A$ lies on the curve $y = x^{2} + 5 x + 8$ Th…	Find the coordinates of point A: (-4, 4)
None	Find the equation of the tangent line to the func…	Find the derivative of the function: $f^{'} (x) = - 18 x + 3$
None	Fid the value of $k$ such that the tangent li…	$f^{'} (x) = \frac{- 2 (k + x)}{x^{3}}$
None	Find the equation of the tangent line to \( g(x)=…	$g^{'} (x) = 3 x^{2}$
None	Find the equation of the tangent line to \( g(x)=…	Find $g^{'} (x)$ by taking the derivative of $g (x) = x^{2}$ .
None	10. Calculate the tangent planes of the following…	\[ \frac{\partial f}{\partial x} = 2x, \frac{\partial f}{\partial y} = 2y \Rightarrow \frac{\partia…