Basic Mathematics · Unit 4
Derivatives and Differentiation
Exam-focused notes for Derivatives and Differentiation (Basic Mathematics, MTH104): what the TU syllabus asks and how it has actually been tested, with 7 solved past questions from this unit.
What this unit covers
- Derivative definition and interpretation
- Derivative as slope of tangent line
- Power rule and basic differentiation rules
- Product and quotient rules
- Chain rule
- Implicit differentiation
- Derivatives of trigonometric functions
- Derivatives of inverse trigonometric functions
- Derivatives of exponential and logarithmic functions
- Higher order derivatives
- Partial derivatives
Derivatives of inverse trigonometric functions
Find the derivative of $y = \frac{\tan^{-1} x}{\sqrt{x}}$ with respect to $x$.
Find the area of the region bounded by $y = -x$ and $x = y^2 + 3y$.
[5+5]
Part 1: $y = \dfrac{\tan^{-1} x}{\sqrt{x}}$ Part 2: Curves $y = -x$ and $x = y^2 + 3y$ --- (a) Derivative Let $u = \tan^{-1} x$ and $v = \sqrt{x} = x^{1/2}$. $$\frac{du}{dx} = \frac{1}{1+x^2}, \qquad \frac{dv}{dx} = \frac{1}{2\sqrt{x}}$$ Quotient rule: $\df...
Full solved answer →Implicit differentiation
Define implicit differentiation and find the slope of the circle $x^2 + y^2 = 25$ at the point (3, -4). [5]
Given data: - Curve (implicit relation): $x^2 + y^2 = 25$ (a circle of radius 5 centered at origin) - Point of interest: $(3, -4)$ Check that the point lies on the circle: $3^2 + (-4)^2 = 9 + 16 = 25$ ✓ All required data present. Implicit differentiation is...
Full solved answer →Find $\frac{dy}{dx}$ of the following. $y^2 - x^2 = \cos(xy)$, $x^2 = \frac{9}{y^2}$. [5]
- Equation 1: $y^2 - x^2 = \cos(xy)$ - Equation 2: $x^2 = \dfrac{9}{y^2}$ Find $\dfrac{dy}{dx}$ for each (implicit differentiation). --- Differentiate both sides with respect to $x$: $$\frac{d}{dx}(y^2 - x^2) = \frac{d}{dx}[\cos(xy)]$$ Left side: $$2y\frac{...
Full solved answer →Find $\frac{dy}{dx}$ if $y^2 - x^2 = \sin x$. Find the slope of circle $x^2 + y^2 = 25$ at the point $(3,4)$. [2.5+2.5]
- Part 1: $y^2 - x^2 = \sin x$ - Part 2: Circle $x^2 + y^2 = 25$, point $(3, 4)$ --- (a) Find $\frac{dy}{dx}$ if $y^2 - x^2 = \sin x$ Differentiate both sides with respect to $x$: $$\frac{d}{dx}(y^2 - x^2) = \frac{d}{dx}(\sin x)$$ $$2y\frac{dy}{dx} - 2x = \...
Full solved answer →Chain rule
Find $\frac{dy}{dx}$ for:
- $Y = 2u^3$, $u = 8x - 1$
- $Y = \sin u$, $u = x - x\cos x$
[5]
Problem 1: $Y = 2u^3$, $u = 8x - 1$ Problem 2: $Y = \sin u$, $u = x - x\cos x$ Chain rule: $\dfrac{dy}{dx} = \dfrac{dy}{du} \cdot \dfrac{du}{dx}$ --- Step 1: Differentiate $Y$ with respect to $u$ $$\frac{dy}{du} = 2 \cdot 3u^2 = 6u^2$$ Step 2: Differentiate...
Full solved answer →Use the chain rule to find $\frac{dz}{dt}$ when $z = \cos(x + 4y)$, $x = 5t^4$, $y = \frac{1}{t}$. [5]
- $z = \cos(x + 4y)$ - $x = 5t^4$ - $y = \dfrac{1}{t}$ Required: $\dfrac{dz}{dt}$ All data present. --- $$\frac{dz}{dt} = \frac{\partial z}{\partial x}\cdot\frac{dx}{dt} + \frac{\partial z}{\partial y}\cdot\frac{dy}{dt}$$ $$\frac{\partial z}{\partial x} = -...
Full solved answer →Derivative as slope of tangent line
Find the slope of the curve $y = 1/x$ at any point $x = a$, $a \neq 0$. What is the slope at the point $x = -1$? Where does the slope equal $-1/4$? What happens to the tangent to the curve at the point $(a, 1/a)$ as $a$ changes? [2+1.5+1.5]
- Curve: $y = \dfrac{1}{x}$ - Point of interest: $x = a$, with $a \neq 0$ - Specific values to evaluate: slope at $x = -1$; where slope $= -\dfrac{1}{4}$; behavior of tangent at $(a, 1/a)$ as $a$ varies --- Write $y = x^{-1}$. Differentiating using the powe...
Full solved answer →Make Unit 4 stick
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