High School
SAT
ACT
University
MCAT
LSAT
DAT
Wize University Calculus 1 Textbook > Derivatives

Implicit Differentiation

Higher Order DerivativesPrevious Section
Derivatives of Inverse FunctionsNext Section
Popular Courses
Find My Course
0:00 / 0:00

Implicit Differentiation

We call a traditional function y=f(x)y=f(x) explicit since yy is in terms of xx. However, even if yy cannot be written explicitly in terms of xx, we can still compute its derivative by differentiating both sides of the equation.


Watch Out!
You must use implicit differentiation to take the derivative of equations that cannot be solved for yy explicitly (or would be very challenging to do so).

Procedure for Implicit Differentiation

  1. Differentiate on both sides of the equation by considering yy as a function of x.x.
  2. Use the chain rule for all yy dependent terms.
  3. Isolate dydx.\dfrac{dy}{dx}. (or y′y'depending on your preferred notation)

Wize Tip
Whenever you take the derivative of a yy , you must multiply the term by dydx\displaystyle \frac{dy}{dx} (or y′y'depending on your notation) due to the chain rule.
0:00 / 0:00

Example: Implicit Differentiation

Find dydx\displaystyle \frac{dy}{dx} for the following equation:

ycos⁡x=x2+cos⁡yy\cos x=x^2+\cos y

Since we can't isolate for yy, we need to use implicit differentiation to find the derivative:
dydxcos⁡(x)+y(−sin⁡(x))=2x−sin⁡(y)dydx\begin{array}{rcl} \displaystyle \dfrac{dy}{dx}\cos(x)+y(-\sin(x)) &=& 2x-\sin(y)\dfrac{dy}{dx} \end{array}

Now solve for dydx\frac{dy}{dx}:
dydx(cos⁡(x)+sin⁡(y))=2x+ysin⁡(x)dydx=2x+ysin⁡(x)cos⁡(x)+sin⁡(y)\begin{array}{rcl} \displaystyle \dfrac{dy}{dx}(\cos(x)+\sin(y))& =& 2x+y\sin(x)\\ \displaystyle \dfrac{dy}{dx} &=& \dfrac{2x+y\sin(x)}{\cos(x)+\sin(y)} \end{array}
0:00 / 0:00

Example: Implicit Differentiation (with 2nd Derivatives)

Find the second derivative, d2ydx2 \displaystyle \frac{d^2y}{dx^2} for the the implicit function:y2x=10\displaystyle \frac{y^2}{x}=10

First we will need to find the first derivative dydx\dfrac{dy}{dx}. (you may also use the y′y'notation)
y2x=10⇒x(2y×dydx)−y2(1)x2=0Note in the original expression, x≠0⇒2xy×dydx−y2=0⇒2xy×dydx=y2⇒dydx=y2x\begin{array}{l} \displaystyle \frac{y^2}{x}=10 \\ \\ \Rightarrow \displaystyle \frac{x(2y\times\dfrac{dy}{dx})-y^2(1)}{x^2}=0 \\ \\ \text{Note in the original expression, } x\ne0 \\ \\ \Rightarrow 2xy \times \dfrac{dy}{dx}-y^2=0 \\ \\ \Rightarrow 2xy \times \dfrac{dy}{dx}=y^2 \\ \\ \Rightarrow \dfrac{dy}{dx}=\dfrac{y}{2x} \end{array}

Now to find the second derivative, take the derivative of dydx\dfrac{dy}{dx}.

dydx=y2x⇒d2ydx2=2x(dydx)−y(2)4x2substitute: dydx=y2x⇒d2ydx2=2x(y2x)−2y4x2d2ydx2=y−2y4x2d2ydx2=−y4x2\begin{array}{l} \dfrac{dy}{dx}=\dfrac{y}{2x} \\ \\ \Rightarrow \displaystyle \frac{d^2y}{dx^2} = \dfrac{2x\left(\dfrac{dy}{dx}\right)-y(2)}{4x^2} \\ \\ \text{substitute: } \dfrac{dy}{dx}=\dfrac{y}{2x} \\ \\ \Rightarrow \displaystyle \frac{d^2y}{dx^2} = \dfrac{2x\left(\dfrac{y}{2x}\right)-2y}{4x^2} \\ \\ \displaystyle \frac{d^2y}{dx^2} = \dfrac{y-2y}{4x^2} \\ \\ \displaystyle \boxed{\frac{d^2y}{dx^2} = -\dfrac{y}{4x^2}} \end{array}

Find dydx\displaystyle\frac{dy}{dx} for the following equation:

tan⁡(yx)=x\displaystyle\tan\left(\frac{y}{x}\right)=x


Extra Practice
Implicit differentiation
Differentiate exy−tan⁡y=3x3+2xe^{xy}-\tan y=\frac{3}{x^3}+2x
Implicit Differentiation: Tangent line
Find the equation of the tangent to the graph 3y3+x=x2+13y^3+x=x^2+1 at x=2x=2.
Tangent lines: Implicit Differentiation
An equation of the tangent line to the curve x3+xln⁡y+y4=3x32+exx^3+x\ln y+y^4=3x^{\frac{3}{2}}+e^x at x=0x=0 is
Implicit Differentiation: Logarithmic Differentiation
Find the equation of the tangent line to
(sin⁡(xy))x=x1/4(\sin(xy))^x = x^{1/4}
at the point (12,π2)\left(\dfrac{1}{2},\dfrac{\pi}{2}\right).
Implicit Second Derivative
Find d2ydx2\dfrac{d^2y}{dx^2} at the point (2,2)(2,2) given xy+y2=2xy+y^2=2
Implicit differentiation
Differentiate exy−tan⁡y=3x3+2xe^{xy}-\tan y=\frac{3}{x^3}+2x
Implicit Differentiation
Find the slope of the line tangent to the curve xy=x3y−6\sqrt{xy}=x^3y-6 at the point (1,9).
Tangent lines: Implicit Differentiation
An equation of the tangent line to the curve x3+xln⁡y+y4=3x32+exx^3+x\ln y+y^4=3x^{\frac{3}{2}}+e^x at x=0x=0 is
Implicit Differentiation
Find y'' if x3+y3=4.x^3+y^3=4.
Implicit Differentiation
Find dydx\displaystyle \frac{\text{d}y}{\text{d}x} for y3=2xy^3=2x.
Implicit Differentiation
Find the slope of the line tangent to the curve xy=x3y−6\sqrt{xy}=x^3y-6 at the point (1,9).
Implicit differentiation
Find the derivative of the following:
yex+y2=x2ye^{x+y^2}=x^2
Tangent lines: Implicit Differentiation
An equation of the tangent line to the curve x3+xln⁡y+y4=3x32+exx^3+x\ln y+y^4=3x^{\frac{3}{2}}+e^x at x=0x=0 is
Find the tangent to the curve 4𝑦2 − 𝑒𝑥 + 𝑥 = 4𝑥 + 3 at the point where 𝑥 = 0 and 𝑦 is positive
Implicit Differentiation
Evaluate the slope of the line tangent to the curve xy=x3y−6\sqrt{xy}=x^3y-6 at the point P(1,9)P(1,9).
Implicit Differentiation: Tangent line
Find the equation of the tangent to the graph 3y3+x=x2+13y^3+x=x^2+1 at x=2x=2.
Implicit Differentiation: Tangent line
Find the equation of the tangent to the graph 3y3+x=x2+13y^3+x=x^2+1 at x=2x=2.
Implicit Differentiation
Evaluate the slope of the line tangent to the curve xy=x3y−6\sqrt{xy}=x^3y-6 at the point P(1,9)P(1,9).
Implicit Differentiation
For the implicit equation x4+x3y−xy2=1x^4+x^3y-xy^2=1, where y = f(x), find y'.
Implicit Differentiation: Tangent line
Find the equation of the tangent to the graph 3y3+x=x2+13y^3+x=x^2+1 at x=2x=2.
Implicit Differentiation
Find all the equations of the tangent lines to the curve y=x3+x2−xy=x^3+x^2-x that are parallel to the x-axis.
Find dydx\frac{dy}{dx} at x=0x=0 if xcos⁡(y2)=ycos⁡xx\cos\left(\frac{y}{2}\right)=y\cos x.
Implicit Differentiation: Tangent line
Find the equation of the tangent to the graph 3y3+x=x2+13y^3+x=x^2+1 at x=2x=2.
Implicit differentiation
Differentiate exy−tan⁡y=3x3+2xe^{xy}-\tan y=\frac{3}{x^3}+2x
Implicit Differentiation
Given x2+y=y2−4x^2+y=y^2-4, find y′y'.
Implicit Differentiation
Find dydx\displaystyle \frac{\text{d}y}{\text{d}x} for y3=2xy^3=2x.
Find the point(s) where x2−4x+6y+y2=−9 x^2-4x+6y+y^2 = −9 has horizontal and vertical tangent lines.
Find dydx\dfrac{dy}{dx} for the function ycos⁡x=x2+cos⁡yy\cos x=x^2+\cos y
Logarithmic for Implicitly Defined Function
Find dy/dxdy/dx at the point (1,1)(1,1) given xy=yxx^y=y^x
Practice: Tangent and Normal Lines with Implicit
Q:\textbf{Q:} Find the equation of the tangent and the normal lines to the curve x2cos⁡2y−sin⁡y=0x^2\cos^2y-\sin y=0 at the point (0,π)(0,\pi).
Implicit Second Derivative
Find d2ydx2\dfrac{d^2y}{dx^2} at the point (2,2)(2,2) given xy+y2=2xy+y^2=2
Implicit Differentiation
Evaluate the slope of the line tangent to the curve xy=x3y−6\sqrt{xy}=x^3y-6 at the point P(1,9)P(1,9).
Implicit with Trig
Find dy/dxdy/dx for xy=tan⁡(x+y2)\displaystyle xy=\tan(x+y^{2})
Implicit Differentiation
Let xx, yy satisfy the equation xy=6y−x+1x^y = 6y - x + 1. Find dydx\frac{dy}{dx} at the point (x,y)=(1,1/6)(x, y) = (1, 1/6).
Derivatives: Logarithmic Functions
Compute the derivative of f(x)=xx+1f(x) = x^{x + 1}. Remember that log⁡x=log⁡ex=ln⁡x\log x = \log_e x = \ln x.
Tangent lines: Implicit Differentiation
An equation of the tangent line to the curve x3+xln⁡y+y4=3x32+exx^3+x\ln y+y^4=3x^{\frac{3}{2}}+e^x at x=0x=0 is
Implicit Differentiation: Logarithmic Functions
For the following equations, solve for dydx\frac{dy}{dx} :
a) y=(5x2+3)sin⁡(x)y = (5x^2 + 3)^{\sin(x)}
b) x2y+3y2x2=x+yx^2y + 3y^2 x^2 = x + y
Evaluate the slope of the tangent line to the curve at the given point.
xy=x3y−6  at  P(1,9)\displaystyle \sqrt{xy}=x^3y-6\,\, at \,\,P(1,9)
Implicit differentiation
Find the derivative of the following:
yex+y2=x2ye^{x+y^2}=x^2
Implicit differentiation
Find the derivative of the following:
sin⁡(x+y)−sec⁡(x2+y2)=y\sin(x+y)-\mathrm{\sec}(x^2+y^2)=y
Implicit Differentiation
Evaluate the slope of the tangent line to the curve at the given point.
x3+y3=ysin⁡x+1,  at  P(0,1)x^3+y^3=y\sin{x}+1,\,\, \text{at}\,\,P(0,1)
Evaluate the slope of the tangent line to the curve at the given point.
xy=x3y−6  at  P(1,9)\displaystyle \sqrt{xy}=x^3y-6\,\, at \,\,P(1,9)
Evaluate the slope of the tangent line to the curve at the given point.
xy=x3y−6  at  P(1,9)\displaystyle \sqrt{xy}=x^3y-6\,\, at \,\,P(1,9)
Find the equation of the tangent line to the equation x3+ln⁡y=8x^3+\ln y=8 at the point (2, 1)
Evaluate the slope of the tangent line to the curve at the given point.
xy=x3y−6  at  P(1,9)\displaystyle \sqrt{xy}=x^3y-6\,\, at \,\,P(1,9)
Implicit Differentiation: Logarithmic Differentiation
Find the equation of the tangent line to
(sin⁡(xy))x=x1/4(\sin(xy))^x = x^{1/4}
at the point (12,π2)\left(\dfrac{1}{2},\dfrac{\pi}{2}\right).
Implicit Differentiation
Find the slope of the line tangent to the curve xy=x3y−6\sqrt{xy}=x^3y-6 at the point (1,9).
Implicit Differentiation
Find y'' if x3+y3=4.x^3+y^3=4.
Find dydx\frac{dy}{dx} given that exy−xy=2x2+1 e^{xy}-xy=2x^2+1
final114
Find y′y' for the function x4+x3y−xy2=1x^4+x^3y-xy^2=1
Differentiate the following functions.
(a) g(t)=(t5+t2)6tg(t)=(t^5+t^2)^{6t} (b) f(x)=x6xf(x)=\sqrt x^{6x} (c) f(x)=(x3+4)tan⁡xf(x)=(x^3+4)^{\tan x} (d) Finddydx\frac{dy}{dx} if ycos⁡(x)=xtan⁡(y)y\cos(x)=x\tan(y)
Differentiate y with respect to x in the equation 2x2+3xy−y2=12x^2+3xy-y^2=1.
Find dydx\frac{dy}{dx} given that exy−xy=2x2+1 e^{xy}-xy=2x^2+1
Evaluate the slope of the tangent line to the curve at the given point.
x3+y3=4xy,  at  P(2,2)x^3+y^3=4xy ,\,\, \text{at} \,\, P(2,2)
Evaluate the slope of the tangent line to the curve at the given point.
x3+y3=ysin⁡x+1,  at  P(0,1)x^3+y^3=y\sin{x}+1,\,\, \text{at}\,\,P(0,1)
Implicit Differentiation
Given x2+y=y2−4x^2+y=y^2-4, find y′y'.
Implicit Differentiation
Find dydx\displaystyle \frac{\text{d}y}{\text{d}x} for y3=2xy^3=2x.
Find the point(s) where x2−4x+6y+y2=−9 x^2-4x+6y+y^2 = −9 has horizontal and vertical tangent lines.
Find dydx\dfrac{dy}{dx} for the function ycos⁡x=x2+cos⁡yy\cos x=x^2+\cos y
Logarithmic for Implicitly Defined Function
Find dy/dxdy/dx at the point (1,1)(1,1) given xy=yxx^y=y^x
Implicit Second Derivative
Find d2ydx2\dfrac{d^2y}{dx^2} at the point (2,2)(2,2) given xy+y2=2xy+y^2=2
Practice: Tangent and Normal Lines with Implicit
Q:\textbf{Q:} Find the equation of the tangent and the normal lines to the curve x2cos⁡2y−sin⁡y=0x^2\cos^2y-\sin y=0 at the point (0,π)(0,\pi).
Implicit Differentiation
Evaluate the slope of the line tangent to the curve xy=x3y−6\sqrt{xy}=x^3y-6 at the point P(1,9)P(1,9).
Implicit with Trig
Find dy/dxdy/dx for xy=tan⁡(x+y2)\displaystyle xy=\tan(x+y^{2})
Find the slope of the tangent line to the curve ln⁡(x−y)+3y2=3\ln(x-y)+3y^2=3 at point (3,2)(3,2).
Find y′y' for the function x4+x3y−xy2=1x^4+x^3y-xy^2=1
Implicit Differentiation: Estimate of Implicit Function
Use a linear approximation to estimate the yy value of the curve described by x3+3xy+y3=5x^3+3xy+y^3=5 at the point for which x=0.1x=0.1 (leave it as an exact value if you're not allowed a calculator on the exam).
Find the point on the graph y2 = 4x + 5 where the tangent line is parallel to the line y = 2x + 184.
Concept Clarifier
Compute the second derivative of the implicit function y3=xcos⁡yy^3=x \cos y, where yy is a function of xx. Do not simplify.
Find the point(s) where x2−4x+6y+y2 = −9 has horizontal and vertical tangent lines.
Find the slope of the tangent line to the curve ln(x − y) + y2 = 4 at point (3,2).
(Express your answer as a fraction in lowest terms. If the answer is a negative number, put the negative sign in front of the entire fraction.)
Find the equation of the tangent line at the point (1,2)(1,2) of the curve y3−2x5y2=yln⁡(x)y^3-2x^5y^2=y\ln(x).
Find the derivative of sin⁡(x+y)−sec(x2+y2)=y\sin(x+y)-\mathrm{sec} (x^2+y^2)=y.
Find the derivative of y with respect to x of the following equation, y2+x5y3=5−ln⁡yy^2+x^5y^3=5-\ln y.
Given that x2+y=y2−4x^2+y=y^2-4, where y is dependent on x, find dydx\frac{dy}{dx}.
Implicit Differentiation
For the implicit equation x4+x3y−xy2=1x^4+x^3y-xy^2=1, where y = f(x), find y'.
Evaluate the slope of the tangent line to the curve at the given point.
xy=x3y−6  at  P(1,9)\displaystyle \sqrt{xy}=x^3y-6\,\, at \,\,P(1,9)
Find the tangent to the curve 4𝑦2 − 𝑒𝑥 + 𝑥 = 4𝑥 + 3 at the point where 𝑥 = 0 and 𝑦 is positive
Implicit Differentiation
Find all the equations of the tangent lines to the curve y=x3+x2−xy=x^3+x^2-x that are parallel to the x-axis.
Find dydx\frac{dy}{dx} at x=0x=0 if xcos⁡(y2)=ycos⁡xx\cos\left(\frac{y}{2}\right)=y\cos x.
Implicit Differentiation: Tangent line
Find the equation of the tangent to the graph 3y3+x=x2+13y^3+x=x^2+1 at x=2x=2.
Implicit differentiation
Differentiate exy−tan⁡y=3x3+2xe^{xy}-\tan y=\frac{3}{x^3}+2x
Implicit differentiation
Suppose 3x3+4y2=193x^3+4y^2=19, where x, yx,\ y are both functions of tt.
If dydt=3\frac{dy}{dt}=3, find dxdt\frac{dx}{dt} when y=2y=2.
Solve for dydx\frac{dy}{dx} :
x2y+3y2x2=x+yx^2y+3y^2x^2=x+y
Implicit Differentiation
Use implicit differentiation to calculate the derivative of
θ(x)=arctan⁡(x)\theta(x) = \arctan(x)
Implicit Differentiation
Find dydx\displaystyle \frac{\text{d}y}{\text{d}x} for ln⁡(x2+y2)+2xy=4\ln(x^2+y^2)+2xy=4.
Implicit Differentiation
Evaluate the slope of the tangent line to the curve at the given point.
x3y2+xtan⁡y=4 at P(x,y)\displaystyle x^3y^2+x\tan{y}=4 \, at\, P(x,y)
Evaluate the slope of the tangent line to the curve at the given point.
x2y3+xln⁡y+ysin⁡x=10  at  P(x,y)\displaystyle x^2y^3+x\ln{y}+y\sin{x}=10\,\, at \,\,P(x,y)
Practice: Implicit Differentiation
Find dydx\frac{dy}{dx} for the following
tan⁡yx=x\tan{\frac{y}{x}}=x