Math 16A - Short Calculus
Homework 11
Implicit Differentiation
Instructions: Show all steps in the solution. Compute the indicated derivatives of the following
functions.
1. First rewrite the equation in explicit form, then find
(a) x3 y = 1
(b) 5x5 y = 2
(c) (x + 1)y = 1
y
=2
(d) 2
x +7
y
(e)
=4
sin(x2 )
2. Differentiate each expression with respect to x.
(a) 3x3
(b) 2y 2
(c) x + 7y
(d) e2x − e2y
(e) 7x2 − 4y 3 +
√
y
2
(f) 8t − 3t + x
(g) ,8
2
(h) ,2 − 4,
(i) e−x + sin(t) − cos(y) + e4z
(j) x2 y 3
(k) sin(x) cos(y)
(l)
tan(2x)
ey
3. For each of the following expressions, compute
dy
.
dx
(a) y = x3 − x
(b) y 2 = sin(x)
(c) ey = tan(x)
(d) y 2 + y = x1/3 − ex
(e) tan(y) = cot(x) − 7
(f) xy = 17x − 4x4
(g) x2 y 5 + 2x = sin(−x)
(h) sec(y) − ey = cos(x2 )
√
(i) sin(x2 y) = 10x2 − x
(j) y = 2x3 y 5 + y 3 x4
√
(k) y − ex y 2 = exy − 4
1
dy
.
dx
4. Below is the graph of the elliptic curve y 2 = x3 − 2x + 1.
(a) Algebraically confirm that (0, 1) is a point on the graph.
(b) Find the equation of the tangent line at the point (0, 1).
(c) Sketch the tangent line on the graph.
5. Below is the graph of the elliptic curve y 2 = x3 − 2x + 2.
(a) Algebraically confirm that (0, 2) is a point on the graph.
(b) Find the equation of the tangent line at the point (0, 2).
(c) Sketch the tangent line on the graph.
6. Below is the graph of the elliptic curve y 2 = x3 .
(a) Algebraically confirm that (1, 1) is a point on the graph.
(b) Find the equation of the tangent line at the point (1, 1).
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7. Below is the graph of the elliptic curve y 2 = x3 + 1.
(a) Algebraically confirm that (−1, 0) is a point on the graph.
(b) Draw the tangent line at (−1, 0) on the graph .
(c) Compute the slope of the tangent line at the point (−1, 0).
(d) Even though the answer to part (c) looks weird, why does it make sense with what you drew for
(b)?
8. Below is the graph of the ellipse x2 + 4y 2 = 16. Find the equation of the tangent line at x = 2 in the
first quadrant and sketch it on the graph.
9. Below is the graph of the hyperbola 3x2 − 2y 2 = 1. Find the equation of the tangent line at x = 2 in
the first quadrant.
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10. Below is the graph of the hyperbola 4x2 − 3y 2 = 1.
(a) Verify that the point (1, 1) is on the hyperbola.
(b) Find the equation of the tangent line (in red) at the point (1, 1).
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