Calculus Examples

Find the Tangent Line at the Point y=(7x)/(x+4) , (3,3)
y=7xx+4 , (3,3)
Step 1
Find the first derivative and evaluate at x=3 and y=3 to find the slope of the tangent line.
Tap for more steps...
Step 1.1
Since 7 is constant with respect to x, the derivative of 7xx+4 with respect to x is 7ddx[xx+4].
7ddx[xx+4]
Step 1.2
Differentiate using the Quotient Rule which states that ddx[f(x)g(x)] is g(x)ddx[f(x)]-f(x)ddx[g(x)]g(x)2 where f(x)=x and g(x)=x+4.
7(x+4)ddx[x]-xddx[x+4](x+4)2
Step 1.3
Differentiate.
Tap for more steps...
Step 1.3.1
Differentiate using the Power Rule which states that ddx[xn] is nxn-1 where n=1.
7(x+4)1-xddx[x+4](x+4)2
Step 1.3.2
Multiply x+4 by 1.
7x+4-xddx[x+4](x+4)2
Step 1.3.3
By the Sum Rule, the derivative of x+4 with respect to x is ddx[x]+ddx[4].
7x+4-x(ddx[x]+ddx[4])(x+4)2
Step 1.3.4
Differentiate using the Power Rule which states that ddx[xn] is nxn-1 where n=1.
7x+4-x(1+ddx[4])(x+4)2
Step 1.3.5
Since 4 is constant with respect to x, the derivative of 4 with respect to x is 0.
7x+4-x(1+0)(x+4)2
Step 1.3.6
Simplify terms.
Tap for more steps...
Step 1.3.6.1
Add 1 and 0.
7x+4-x1(x+4)2
Step 1.3.6.2
Multiply -1 by 1.
7x+4-x(x+4)2
Step 1.3.6.3
Subtract x from x.
70+4(x+4)2
Step 1.3.6.4
Add 0 and 4.
74(x+4)2
Step 1.3.6.5
Combine 7 and 4(x+4)2.
74(x+4)2
Step 1.3.6.6
Multiply 7 by 4.
28(x+4)2
28(x+4)2
28(x+4)2
Step 1.4
Evaluate the derivative at x=3.
28((3)+4)2
Step 1.5
Simplify.
Tap for more steps...
Step 1.5.1
Simplify the denominator.
Tap for more steps...
Step 1.5.1.1
Add 3 and 4.
2872
Step 1.5.1.2
Raise 7 to the power of 2.
2849
2849
Step 1.5.2
Cancel the common factor of 28 and 49.
Tap for more steps...
Step 1.5.2.1
Factor 7 out of 28.
7(4)49
Step 1.5.2.2
Cancel the common factors.
Tap for more steps...
Step 1.5.2.2.1
Factor 7 out of 49.
7477
Step 1.5.2.2.2
Cancel the common factor.
7477
Step 1.5.2.2.3
Rewrite the expression.
47
47
47
47
47
Step 2
Plug the slope and point values into the point-slope formula and solve for y.
Tap for more steps...
Step 2.1
Use the slope 47 and a given point (3,3) to substitute for x1 and y1 in the point-slope form y-y1=m(x-x1), which is derived from the slope equation m=y2-y1x2-x1.
y-(3)=47(x-(3))
Step 2.2
Simplify the equation and keep it in point-slope form.
y-3=47(x-3)
Step 2.3
Solve for y.
Tap for more steps...
Step 2.3.1
Simplify 47(x-3).
Tap for more steps...
Step 2.3.1.1
Rewrite.
y-3=0+0+47(x-3)
Step 2.3.1.2
Simplify by adding zeros.
y-3=47(x-3)
Step 2.3.1.3
Apply the distributive property.
y-3=47x+47-3
Step 2.3.1.4
Combine 47 and x.
y-3=4x7+47-3
Step 2.3.1.5
Multiply 47-3.
Tap for more steps...
Step 2.3.1.5.1
Combine 47 and -3.
y-3=4x7+4-37
Step 2.3.1.5.2
Multiply 4 by -3.
y-3=4x7+-127
y-3=4x7+-127
Step 2.3.1.6
Move the negative in front of the fraction.
y-3=4x7-127
y-3=4x7-127
Step 2.3.2
Move all terms not containing y to the right side of the equation.
Tap for more steps...
Step 2.3.2.1
Add 3 to both sides of the equation.
y=4x7-127+3
Step 2.3.2.2
To write 3 as a fraction with a common denominator, multiply by 77.
y=4x7-127+377
Step 2.3.2.3
Combine 3 and 77.
y=4x7-127+377
Step 2.3.2.4
Combine the numerators over the common denominator.
y=4x7+-12+377
Step 2.3.2.5
Simplify the numerator.
Tap for more steps...
Step 2.3.2.5.1
Multiply 3 by 7.
y=4x7+-12+217
Step 2.3.2.5.2
Add -12 and 21.
y=4x7+97
y=4x7+97
y=4x7+97
Step 2.3.3
Reorder terms.
y=47x+97
y=47x+97
y=47x+97
Step 3
 [x2  12  π  xdx ]