47-56. Derivatives of inverse functions at a point Consider the following functions. In each case, without finding the inverse, evaluate the derivative of the inverse at the given point. 56. f (x) log10 3 x; 0,
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- (a) Find the derivative. y = ln(4x − 2) − 3 ln(x) dy dx = (b) Find the derivative. y = ln 4x − 2 x3 dy dx = State whether the function in part (b) is the same function as that in part (a). The function in part (b) is the same function as that in part (a).The function in part (b) is not the same function as that in part (a).Using Leibniz find the third derivative of the following function y=x2cos3x Find the local linear approximation L(x) of the function f(x)=lnx at x=1 .Find the derivative of the function.5. y = log (9x - 7)6. y = log |6 - x| Use logarithmic differentiation to find the derivative of y.7. y = (x^3+ 1)^3 (x - 1)^5 x^3 Find the derivative of y with respect to x.8. y = cos^-1 (11x^2 + 4)9. y = sin^-1 (e^3t)
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- Differentials Consider the following function and expressthe relationship between a small change in x and the correspondingchange in y in the form dy = ƒ'(x)dx. ƒ(x) = ln (1 - x)Waste material from a mining operation is dumped on a “slag-heap”, which is a large mound, roughly conical in shape, which continually increases in size as more waste material is added to the top. In a mathematical model, the rate at which the height h of the slag-heap increases is inversely proportional to h2.I. Express this statement as a differential equation relating h with time t. II. Show by integration that the general solution of the differential equation relating h and t may be expressed in the form h3 = At + B, where A and B are constants. III. A new slag- heap was started at time t = 0, and after 2 years its height was 18m. Find the time by which its height would grow to 30m.Question: Find the first derivative of y with respect to x when x=2: y=ln (x^2/sin2x) Answer: -0.727 Explain the process and methods.
- SUBJECT: DIFFERENTIAL CALCULUS FIND THE FIRST DERIVATIVE OF THE FOLLOWING FUNCTIONS. SIMPLIFY YOUR ANSWERS. 3. y = (x – 1) √ (2x – x2) + arcsin (x – 1) 6. y = ln [ x + √ (x2+ 1) ] NOTE: Make the solution detailed and clear in a sheet of paper.Differentials Consider the following function and expressthe relationship between a small change in x and the correspondingchange in y in the form dy = ƒ'(x)dx. ƒ(x) = e2xVerify that the function satisfies the differential equation. (A differential equation in x and y is an equation that involves x, y, and derivatives of y.) Function y = 2 sin x + 3 Differential Equation y″ + y = 3