If we draw a graph of a function, and we draw a straight line that just touches the curve at a Point then that point is called tangent. The derivative of tangent is just the differentiation of the function at that particular point. We can find the derivative of the tangent by just differentiating the given function.

Let’s see the some example how to find derivative of a tangent.

Example 1: Find the derivative of tangent at points (-1,1) if x^{2} +4y^{2} +x^{3} +y=0?

Solution: We need to take the derivative of both the side first then, we will integrate that,

d/dx(x^{2}) +d/dx(4y^{2}) +d/dx(x^{3}) + d/dx(y) =0

2x +8y (dy/dx) +3x^{2} +dy/dx =0

Now, we will separate the terms:

8y(dy/dx) +dy/dx =-2x -3x^{2,}

Dy/dx(1+8y) = -2x-3x^{2,}

Dy/dx= -2x-3x^{2}/1+8y,

Now, put the given point to get the required solution:

We will put x=-1 and y=1,

-2(-1) -3((-1)^{2}) / 8*1,

-1/8,

This the required solution

Example 2: Find the derivative of tangent at points (0,0) if x^{2} +4y^{2} +sinx +y=0?

Solution: We need to take the derivative of both the side first then we will integrate that

d/dx(x^{2}) +d/dx(4y^{2}) +d/dx(sinx ) + d/dx(y) =0,

2x +8y (dy/dx) +sinx +dy/dx =0,

Now we will separate the terms,

8y(dy/dx) +dy/dx =-2x -sinx,

Dy/dx(1+8y) = -2x-sinx,

Dy/dx= -2x- sinx /1+8y,

Now, put the given point to get the required solution:

Dy/dx =-2(0)-sin0/1+8(0).

Dy/dx=0,

This is the required solution.

Example 3: Find the derivative of tangent at points (90,0) if ysinx +cosx =0?

Solution: We need to take the derivative of both the side first then we will integrate that,

d/dx(ysinx) + d/dx(cosx),

Now, we need to apply Product rule,

y d/dx(sinx) + d/dx(y) siinx =0,

y* cosx + dy/dx * sinx =0,

dy/dx = -y*cosx /sinx,

Now, put the given point to get the required solution:

Hence y=0 so whole equation will be 0

This is the required solution.

Example 4: Find derivative of tangent at points (2,1) if x^{2} + 4y +4y^{3}=98?

Solution: We need to take the derivative of both the side first then, we will integrate that.

d/dx(x^{2}) + d/dx(4y) + d/dx(4y^{3}) = d/dx(98),

2x + 4* dy/dx + 12y^{2} *dy/dx = 0,

2x + dy/dx (4 + 12y^{2}) =0,

Dy/dx = -2x/4+12y^{2,}

Now put the value of the function we will get:

-2*2/4 +12(1)^{2,}

-1 +12,

11.

This is the required solution.